Numerical modeling, analysis, and simulations for the space-time wave equation with velocity initial and final time conditions

Numerical modeling, analysis, and simulations for the space-time wave equation with velocity initial and final time conditions David Jean Baptiste Bilong ([email protected]) African Institute for Mathematical Sciences (AIMS) Cameroon Supervised by: Prof.Dr. Thomas Wick Leibniz University Hannover, Germany 28 May 2025 Submitted in Partial Fulfillment of a Structured Masters Degree at AIMS-Cameroon Abstract Two variations of the wave equation are examined in this paper with typical beginning circumstances, the first one match the classical wave equation. The second is a recently suggested space-time variational material model where mechanical, thermal, and internal factors interact. It is derived from a thermodynamically consistent Hamilton functional defined across a space time domain. The apparition of a non-standard final time velocity condition in the second order wave equation is a characteristic of this model. We extract and evaluate the wave equation independently, ignoring temperature and internal variables, in order to better grasp the ramifications of the unusual boundary conditions in time. We perform then a Galerkin finite element discretization in time and space for both problem cases. Moreover, we investigate the behavior of these formulations using numerical simulations based on the discretization. Key words: Final time velocity condition, Galerkin finite element discretization, Variational material model, Space-time, Wave equation. Declaration I, the undersigned, hereby declare that the work contained in this essay is my original work, and that any work done by others or by myself previously has been acknowledged and referenced accordingly. DAVID JEAN BAPTISTE BILONG, 28 May 2025. i Contents Abstract i 1 Introduction 1 1.1 Problem Statement ..................................... 1 1.2 Literature Review ...................................... 2 1.3 Thesis Organization ..................................... 3 2 Elastic wave equations: Two perspectives 4 2.1 Classical linear elasto dynamics problem .......................... 4 2.2 Wave equation problem from elastodynamic with final time velocity conditions ..... 5 3 Space-time Galerkin finite element method applied to both models. 17 3.1 Mathematical preliminaries ................................. 17 3.2 Continous Galerkin method applied to the classical linear elastodynamic model . . . . . 18 3.3 Continous Galerkin method applied to our unusual model ................. 22 4 Analysis and Simulations: Results and Discussions 26 4.1 Numerical setup ....................................... 26 4.2 Simulation results ...................................... 27 4.3 Analysis and insights .................................... 30 4.4 Discussion .......................................... 33 Acknowledgements 36 References 37 ii 1. Introduction Over years physicists have been interested by studying complex phenomena implying different variables and this ended sometime with system of equations specifically partial differential equations (PDEs) with derivatives showing the high order of the model,and due to the difficulty for the most of them to be directly solve, numerical methods have then been introduced by mathematicians to try to find an approximate solution having an error approximation as smaller as possible. More specifically in modelisation of complex phenomena implying applied mechanics and thermodynamics and when dealing with elastic body we lead to the so-called elastic wave equation which is a system of non linear PDEs of second order in displacement variable called usually ”u” depending of the space and time with boundaries and initial conditions, the latest usually initial displacement and velocity conditions prescribed to have a well-posed problem. However by analyzing that system of equation more closely, we can remark the imposition of conditions almost everywhere except to the top cover of the space-time cylinder,”cylinder” here is a word use in the field to specify the domain where we work in and having a spatial and temporal dimension. Recently in [11] scientists wishing to study a case of having conditions on the full entire system to see the influence on that on the solution, have got a new variational formulation deriving by using a well construct extended Hamilton functional, and a famous principle from Hamilton,the Hamilton principle, saw the first time in 1836 in his book [8],this is going to expose us at an unusual initial velocity conditions replacing the usual initial displacement condition, a new formulation which is actually part of a whole thermo-mechanical coupled problem implying the reduce set of thermodynamical states {θ,u}, where θis the temperature variable [11]. A first numerical study has been made on that unusual system of wave equation in [12] to study the influence of that new condition by using the so-called Galerkin Finite element method more specifically the continuous one use in time and space which allows as finite element method for more flexibility in the discretization. In this thesis, we aim to add at the first investigation, some new insights by using the same numerical space-time approach (continuous Galerkin finite element method) and more numerical implementations and give some differences raised by doing a comparative study, an exploration on the differences of the error approximations of our results between the classical sense (with usual initial conditions and the new formulation with final time condition added) to get out the important parameters influencing the error approximations in order to understand the influence of our newly conditions. 1.1 Problem Statement This work focuses on the study of our new wave equation system with an unusual final time condition on the velocity. First, we aim to study the approximate solution first of the classical wave equation using the continuous Galerkin finite element method in space and time and also by being interested to the order of the error in order to do a comparison later. Secondly, we will study also the approximate solution and the order of the error for our system with our unusual initial condition. Then, by conducting a comparative analysis on the error approximation, we will compare the approximate solutions coming from the two different posed problems in order to understand and get more insights on the influence of our new condition. 1 Section 1.2. Literature Review Page 2 By analyzing the simulation results, we try to understand the numerical behavior on the error approximations from the different cases. Moreover, we will extract and study the behavior of the important parameters influencing our understanding and give us more insights on how the error changes with the change of parameters numerically in the two cases. 1.2 Literature Review Variational calculus which is the notion used in order to describe physically observed phenomena, specifically coming from applied mechanics have evolved during the years. Many works in that field have been done into the aim trying to describe by variational calculus and energy principles as more as possible the physical phenomena or even exactly how it behaves. Among the methods developed based on energy variations on a body, we have the so-called Hamilton’s principle developed for the first time by Hamilton [8] in 1834 which has given the key to get in some cases the so-called weak variational formulation without going through the strong one with strong imposition of boundary conditions. From that period, with the time the principle has evolved and scientists has started to understand so many other things, specifically the possibility to impose weakly conditions on a model by adding some terms in the Hamilton functional before applying the Hamilton’s principle rather to impose strong conditions on the system. In this regard, a recent work has been done by Philipp Junker and Daniel Balzani [10]. They have showed how to derive evolution equations with state variables by using the Hamilton’s principle and give an example on the use of the extended Hamilton’s principle to derive evolution equations in thermo-elastic materials for small deformations. An interesting direct application of that idea has been the key to construct the extended Hamilton functional, constructed by Thomas Wick and Philipp Junker [11], getting at the end of derivations an unusual final time condition [11]. To study such kind of system of elastic wave equation in space time of second order, we need numerical methods as finite element methods in space and finite difference one in time, this method works but sometime depending of the order of the error with which we want to get at the end, the so-called Galerkin finite element method gives more advantages in the flexibility within the discretization and then on the error. Work on the advantages by theoretical and rigorous proofs have been done in [9] by T.J.R. Hughes and G.M. Hulbert in order to support the idea for using Galerkin finite element method specifically discontinuous there, for this kind of problem (elastic wave equation) and more generally for hyperbolic equations of second order. More specifically for our case a work which has been very useful to understand and have the order of convergence of the continuous Galerkin finite element method applied on the so-called elastic wave equation in the classical sense with usual initial conditions has been developed in [3,7]. So far the study made by all those different scientists from applied mechanics and mathematics for the study of evolution equations described by PDEs and suitable numerical methods as Galerkin finite element method to approximate solutions have played a fundamental role on our understanding of physical phenomena as displacement of particles in an elastic body case. Section 1.3. Thesis Organization Page 3 1.3 Thesis Organization This thesis investigates the numerical modeling and analysis of the classical and unusual wave equations, focusing on the impact of final time velocity condition using the continuous finite element Galerkin method (cG-FEM). The thesis begins with an introduction in Chapter 1, outlining the problem, reviewing relevant literature, and defining the objectives. In Chapter 2, we present the mathematical formulations of both wave equations, by detailing the classical elasto-dynamics model and deriving the unusual model via thermodynamic principles and an extended Hamiltonian approach. After that Chapter 3 follows describing the cG-FEM discretization for both cases, including fully discrete variational problems. Finally, Chapter 4presents numerical simulations conducted on FEniCS, including some analysis, key findings, limitations and future research directions. 2. Elastic wave equations: Two perspectives 2.1 Classical linear elasto dynamics problem 2.1.1 Preliminaries. Here, we consider an open bounded region Ω⊆Rd, with dthe dimension of the space which can take as value 1,2 or 3. We call ∂Ωthe boundary of Ωand which is supposed smooth so that we can define an outward normal vector non it. Furthermore, we assume to have the boundary which can be decomposed as :∂Ω=∂ΩN∪∂ΩDwith ∂ΩNand ∂ΩDare two disjoint sets meaning that ∂ΩN∩∂ΩD=∅. Here, ∂ΩNis in fact where we impose the boundary Newman condition to the system, and ∂ΩDis the one where the Dirichlet condition is imposed. This is the definition of the spatial space. We also consider the interval of time I= [0, T] with T∈]0,+∞[,There is the time length interval of the time domain T. 2.1.2 Problem’s Model. Here, we assume to have an elastic body occupying the region Ωnot necessarily consider homogeneous in which we applied a force fin the body’s volume such that f:Ω−→Rdand a traction forces on the surface’s boundary of Ω(specifically ∂ΩN)gN:∂ΩN−→Rdand a prescribe boundary displacement condition on Dirichlet boundary describe by the function gD:∂ΩD−→Rd. Also we consider the vector displacement u(x, t)depending on space and time describing the evolution of the wave created by the action of the forces in our body. Due to the external forces applied on the body, we have a deformation(strain) of the body and there is a constraint force created by the body to resist to the deformation, the so-called stress which is denoted by σand depend of ∇u and is given by the Hooke’s law which gives the proportional relation between the deformation(strain) and the stress, by the formula: σ(∇u) = c·(u), where:  cis the dilatational wave velocity which changes depending of the material and its nature. For example for the isotropic case(where the properties of the material are the same in all the direction) c is given by c=qλ+2µ ρ,λand µare called Lame parameters, depend of the body and ρthe density of the body ;(see more in [9]). It’s also worth mentioning even though it will not be helpful for us in our developments, that cis actually a fourth order tensor which respect some symmetry properties as: cijkl =cjikl =cijlk = cklij properties that is very helpful to analyze a numerical solution of the wave equation system below by using numerical method as discontinuous Galerkin finite element(see more in [9]).  the strain(deformation) represented by is given by the formula: (u) = 1 2(∇u+∇tu). Under the assumption of small displacements, we have the displacement variable uwhich respect the following system: 4 Section 2.2. Wave equation problem from elastodynamic with final time velocity conditions Page 5            ρ∂2u ∂t2−∇·σ(∇u) = f, on Ω×]0, T[ u=gD, on ∂ΩD×]0, T[ n·σ(∇(u)) = gN, on ∂ΩN×]0, T[ u(x, 0) = u0(x), x ∈Ω ˙u(x, 0) = v0(x), x ∈Ω. (2.1.1) Here, we have the imposition of the two initials conditions to ensure the well-posedness of the problem (i.e., existence,uniqueness of the solution and its continuously depending of the data) since we have a partial differential equation of second order in time. 2.1.3 Example with Ω⊂R.We have in this case σwhich can be written as σ=cu0with c > 0, u0is the spatial derivative and ˙uthe temporal one. The system which can then still be written in this case as follow            ρ¨u−cu00 =f, on Ω×]0, T [ u=gD, on ∂ΩD×]0, T[ cu0=gN, on ∂ΩN×]0, T[ u(x, 0) = u0(x), x ∈Ω ˙u(x, 0) = v0(x), x ∈Ω. (2.1.2) 2.2 Wave equation problem from elastodynamic with final time velocity conditions 2.2.1 Motivation. As we can see previously in the Section 2.1 where we get out the usual model of the elastic wave equation, we raised that the boundary conditions and initial conditions are imposed on the following parts: ∂ΩD×]0, T[,∂ΩN×]0, T[,Ω× {0}, so in fact our conditions cover almost all the system except the top cover of the cylinder. So, a main motivation of this study specifically of this part is to capture the entire surface, define conditions on the entire surface and this is going to lead us at our new perspective of our system with a new condition on time. A derivation of such system has been introduced and demonstrated by Philipp Junker and Thomas Wick [11]. Note.In the following parts we will consider the same preliminaries as in the Section 2.1 . 2.2.2 Thermodynamical states. Considering our bounded region Ωand an elastic body included inside, in each spatial point xwe define or there are properties related to our body, we have two types of material properties: time-dependent material property and time-independent one. In fact timeindependent properties are linked to constant parameters , but here we will be interested on those which are time-dependent usually called Thermodynamic state variables which basically in the thermodynamic is the set having the displacement u, temperature θ, and the internal variables related to the material αfor example it can be the concentration, the plastic strain,the damage state or a combination of them(details can be found in [10]), but for our study of the wave equation for an elastic body we will restrict our study with the thermodynamical state variables which are the displacement and the temperature. S:= {u,θ}. 2.2.3 Remark. Again as recall, our word cylinder that we use here is just an another name to mean our body inside a spatial-time domain. Indeed we can see our spatial domain Ωas a circle through which a temporal dimension passes creating the so-called space-time cylinder. Section 2.2. Wave equation problem from elastodynamic with final time velocity conditions Page 6 2.2.4 Useful laws from thermodynamic. Our final goal on this part will be to give a description of the evolution of thermodynamic state variables with the inclusion of our new condition, but first of all, we have to give some important laws that we need, to be able to conduct a good derivation. More details on these laws can be found in [2,10].  First law of thermodynamic This is the law so-called balance of energy and states that the balance of energy which states the rate of the total energy of the body equals the power due to the mechanical and thermal loads and is translated by the following relation: ˙ E+˙ K=Pmech +Ptherm.(2.2.1) Then, by integrating this equation by time we obtain the equivalence equation below with energies with some constant but we set the constant null and work with it without harming the generality (see more details in [10]). K+E=W+Q.(2.2.2) Integrating equation 2.2.2 over time again, we have the following results ZI Kdt +ZI Edt =ZI Wdt +ZI Qdt, (2.2.3) with: –The kinetic energy Kis given by: K(u)=RΩ 1 2ρkuk2dV –The internal energy Eis given by: E[u, θ]=RΩψ(,θ)dV +RΩθsdV . 2.2.5 Remark. Generally Edepends also of αbut for our case we have considered no internal variables. . –the external mechanical work Wgiven by the expression: W(u)=RΩf·udV +R∂ΩN gN·udA. –external thermal work Qgiven by: Q=RΩRhdtdV −R∂ΩN,θRn·qdtdA where –ρis the density of the body with unit kg/m3. –ψwith unit J/m3is the free energy density, sis the entropy density with unit [s] = J/km3. –The heat flux on the surface qwith unit W/m2,the body force fwith unit N/m3,the external heat source h with unit W/m3,and the traction force gNwith unit N/m2are the quantities given and threaten as constants. –We have which is as in the Section 2.1 the strain or deformation dimensionless and σ=∂ψ ∂ with unit of σwhich is N/m2. –q=−w∇(θ),with wwhich is the heat conductivity and [w] = W/mK Section 2.2. Wave equation problem from elastodynamic with final time velocity conditions Page 13  Construction of the extended Hamilton functional :Here, our goal is to build a functional to which we are going to applied the Hamilton principle that we are going to see below again. In order to derive a weak formulation of our propositional case with final time condition. So, to do that a very careful construction has been chosen (see more details in [11]) in order to apply the Hamilton principle which is described as follows. We consider our definition again on total action, equation 2.2.6 which can again be define as: AI:= 2K − Z∂I ZΩ ρ˙u∗·udV ds. On the other hand, by using the equality given in Proposition 2.2.7 coming actually from the two fundamental laws of thermodynamic, we can then replace in AIone of the two Kby this equality from Proposition 2.2.7. The part containing the expression of AIwithout the value R∂I RΩρ˙u∗·udV ds will be called the principal part of our functional. The word extended will come from the other terms which in fact for a mathematical purpose are additional terms containing all our conditions that we wish to impose weakly and then of course the value −R∂I RΩρ˙u∗·udV ds will enter in those additional terms. The additional terms are defined in what we denote here HAd defined as: HAd[u, θ] := ZIZ∂ΩD,u 1 2cuku−gDk2dAdt −Z∂I ZΩ ρ˙u∗·udV ds +ZIZ∂ΩD,θ 1 2cθ(θ−θ∗)2dAdt +ZΩ 1 2k(θ−θ∗ 0)2dV |t=0. –At t= 0 the expression RΩ 1 2k(θ−θ∗ 0)2dV |t=0 is use to impose the initial temperature θ∗ 0 –The term RIR∂ΩD,u 1 2cuku−gDk2dAdt is used to impose the term of the boundary Dirichlet condition on the displacement and the same for the similar term for temperature. –The term −R∂I RΩρ˙u∗·udV ds is used to impose the initial and end time velocity condition to the system which is our goal, with ∂I ={0, T} –cu,cθare such that:[cu]=Kg/m3s,[cθ]=Kg/msK2. –The heat capacity is denoted by k, with [k] = J/m3k Finally, we can define the extended Hamilton functional that we denote by HEx here: HEx[u, θ] = Hpr[u, θ] + HAd[u, θ]. The term Hpr[u, θ]is given by: Hpr[u, θ] =: ZI (K − gI−ZΩZ˙ θsdtdV −ZΩZ1 θq· ∇θ dtdV −ZΩZ∆sθ dtdV )dt. Note.It worth mentioning to recall that gIis our equation 3.2.5 defining the total potential.  Utilisation of the Hamilton principle :The Hamilton principle is used here from a similar way to what we have seen before with the principle of the least action. Indeed the both in the modern literature are the same and almost used from the same way, but it is worth mentioning Section 2.2. Wave equation problem from elastodynamic with final time velocity conditions Page 14 that historically the principle of least action was dealing with an action with more restriction where the time was not consider as an independent variable ,the Hamilton principle is more general and have with objective to derive the Lagrange equations for the dynamical systems, see more in [8]. So, again using the stationary condition from the Hamiltonian principle as in the principle of least action shown above, we get: δH[u, θ](δu, δθ) = 0 ∀δu, δθ. (2.2.9) Before to continuing to the details, let us see the following proposition. 2.2.12 Proposition. [11,6] Let F:Ω⊂X→Ybe a mapping with X=X1×X2,here u∈X1and θ∈X2, and X, Y are normed vector spaces and Ωopen set. If Fis differentiable at a point a∈Ω,the N(N= 2 here due to u,θ) partial derivatives (∂jF(a)(δaj))1≤j≤Nexist and it holds into all directions δa(δa1, ..., δaN)∈X that: F0[a](δa) = N X j=1 ∂jF(a)(δaj). Proof. see [6], Theorem 7.1-2,p.455 So, using the Proposition 2.2.12, we can still write equation 2.2.9 as: δH[u, θ](δu, δθ) = δuH[u, θ](δu) + δθH[u, θ](δθ) = 0 ∀δu, δθ. 2.2.13 Definition. [11] Let Xand Ybe normed vector spaces again and Ωand open set in X. Here we suppose Y=R , and we consider G:Ω⊂X→Ybe a mapping, b∈Ωand a non zero vector δb∈X. Let assume the parameter ⊂R→ G(b+δb)∈Yis differentiable at = 0. Then, Ghas at b∈Ωthe gˆateau derivative into the direction δb, meaning in fact a directional derivative which is defined by δG[b](δb) := lim →0 G(b+δb)− G(b) ,with δG[b](δb)∈Y. 2.2.14 Proposition. Using the definition above, we get the gateaux derivatives of Hbelow which vanish each one because in the equation just above the variables are independent so to get that equality, we need each gˆateau derivative to vanish. More specifically when δuH[u, θ](δu)=0, we get      RI(RΩ ∂ψ ∂ :δdV −RΩf·δudV −R∂ΩN,u gN·δudA)dt −RIRΩρ˙u·δ˙udV dt −RIR∂ΩD,u cu(u−gD)·δudAdt +R∂I RΩρ˙u∗·δudV ds = 0,∀δu. Proof. Let us show that : δuH[u, θ](δu) = ZI (ZΩ ∂ψ ∂ :δdV −ZΩ f·δudV −Z∂ΩN,u gN·δudA)dt −ZIZΩ ρ˙u·δ˙udV dt −ZIZ∂ΩD,u cu(u−gD)·δudAdt +Z∂I ZΩ ρ˙u∗·δudV ds = 0,∀δu. Section 2.2. Wave equation problem from elastodynamic with final time velocity conditions Page 15 We have : HEx[u, θ] = Hpr[u, θ] + HAd[u, θ] Let us focus on the principal part Hpr and derive its terms one by one, we have: δuK=ZΩ ρ˙u·δ˙u, so δuZI Kdt =ZIZΩ ρ˙u·δ˙udV dt. Moreover, gI=ZΩ ψ(, θ)dV −ZΩ ρ·udV −Z∂ΩN,u gN·udA, where depends on uas in the following expression: =1 2(∇u+∇Tu), δ =1 2(∇δu +∇Tδu). So, δugI=ZΩ ∂ψ ∂ :δdV −ZΩ f·δudV −Z∂ΩN,u gN·δudA. We can remark that all the other terms (thermal ones) do not depend of the variable uso at the end we get: δuHP r =ZI (ZΩ ρ˙u·δ˙udV −ZΩ ∂ψ ∂ :δdV +ZΩ f·δudV +Z∂ΩN,u gN·δudA)dt. For the additional terms included in HAd, by deriving them we get: δuHAd =ZIZ∂ΩD,u cu·(u−gD)·δudAdt −Z∂I ZΩ ρ˙u∗·δudV dt. Finally, we get the equality: δuH[u, θ](δu) = ZI (ZΩ ∂ψ ∂ :δdV −ZΩ f·δudV −Z∂ΩN,u gN·δudA)dt −ZIZΩ ρ˙u·δ˙udV dt −ZIZ∂ΩD,u cu(u−gD)·δudAdt +Z∂I ZΩ ρ˙u∗·δudV ds = 0 ∀δu = 0. 2.2.15 Remark. We have in this equation the direction δu which are admissible directions in a suitable function space. The system is also known as the so-called weak form of the balance of linear momentum for the displacement vector u. Differently from the usual formulation or classical one we have even though it is weakly imposed the initial and final time condition on the velocity. Section 2.2. Wave equation problem from elastodynamic with final time velocity conditions Page 16 Also this equality from the previous Proposition 2.2.14 can still be written as: −ZΩ ρ( ˙u−˙u∗ T)·δ˙udV |t=T+ZΩ (ρ¨u−∇·σ−f)·δudV dt −ZΩ ρ( ˙u−˙u∗ 0)·δudV |t=0 +ZIZ∂ΩN,u (n·σ−gN)·δudAdt +ZIZ∂ΩD,u cu(u−gD)·δudAdt = 0,∀δu. Since using the Cauchy’s law we have that RΩ ∂ψ ∂ :δdV =RΩ∇ · (σ·δu)dV −RΩ∇ · σ·δudV =R∂ΩN,u n·σ·δudAdt −RΩ∇ · σ·δudV . And by integration by parts, we have −ZIZΩ ρ˙u·δ˙udV dt +Z∂I ZΩ ρ˙u∗·δudV ds =−Z∂I ZΩ ρ( ˙u−˙u∗)·δudV ds +ZIZΩ ρ¨u·δudV dt =−ZΩ ρ( ˙u−˙u∗)·δudV |t=T−ZΩ ρ( ˙u−˙u∗)·δudV |t=0 +ZIZΩ ρ¨u·δudV dt. 2.2.16 Remark. So the equation above is the weak variational formulation of our wave elastic system of equation with initial and final time condition. By using the fact that the equality has to be vanished and each of those values are independent regarding to the domain of integration, each one of them has to vanish and we get the system that we call strong formulation given by:            ρ∂2u ∂t2−∇·σ(∇u) = f, on Ω×]0, T[ u=gD, on ∂ΩD×]0, T[ n·σ(∇(u)) = gN, on ∂ΩN×]0, T[ ˙u(x, T) = ˙u∗ T(x), x ∈Ω ˙u(x, 0) = ˙u∗ 0(x), x ∈Ω. (2.2.10) 3. Space-time Galerkin finite element method applied to both models. In this chapter, we first give some mathematical preliminaries as spaces that we deal with to define the discrete spaces necessary for our Continuous Galerkin discretization. Secondly, we applied the spacetime Continuous Galerkin method for the classical wave equation. Finally, we applied for our unusual wave equation system with initial and final time velocity conditions. 3.1 Mathematical preliminaries Here, we give some definitions of the L2space, Sobolev space (more specifically H1(Ω)), also we define what we call by Dual space, we define the euclidean norm in Rdand the so-called time-dependent Sobolev space assuming to have a space Ω⊆Rdwhich is an open bounded region with a sufficient regular boundary. 3.1.1 L2spaces(see in [5]). This space L2is the one of all the measurable functions u: Ω→R(or C) that are square-integrable with respect to the Lebesgue measure define as, L2(Ω) = {u: Ω →R|ZΩ |u(x)|2dx < ∞}. This space is equipped with the inner product, (u, v)L2=ZΩ u(x)v(x)dx, which induces the norm given as kukL2= (ZΩ ku(x)k2)1 2. 3.1.2 Sobolev’s space H1(see in [5]). The Sobolev space H1consists of functions in L2whose the first derivative are also in L2and define on Ωby: H1(Ω) = {u∈L2(Ω) | ∇u∈L2(Ω)}. 3.1.3 Dual space(see in [5]). Let Vbe an Hilbert space(e.g. V=L2(Ω),H1(Ω)) with the norm k.kV, we call the Dual space of Vdenoted by V∗the set of all continuous linear functionals on V. And if we consider a linear functional fin V∗and v∈Vwe define in fact the duality pairing between them by: hf, viV∗,V =f(v). For example, we can consider V=L2(Ω), the set V∗=L2(Ω) and the pairing is hf, viL2,L2=ZΩ f(x)v(x)dx. 17 Section 3.2. Continous Galerkin method applied to the classical linear elastodynamic model Page 18 3.1.4 Time-dependent Sobolev space(see in [5]). We consider our time interval I= [0, T]and V an Hilbert space with norm k.kV; We define the time-dependent Sobolev space L2(I, V )which consists of all measurable functions u:I→Vsuch that ZI ku(t)k2 Vdt < ∞. The inducing norm is given by the expression ku(t)k2 L2(I,V )= (ZI ku(t)k2 Vdt)1 2. 3.1.5 Euclidean norm in Rd(see in [5]). We define the euclidean norm in Rddenoted here as k.kby ∀x= (x1, x2, ..., xd),kuk= ( d X i=1 x2 i)1 2. 3.2 Continous Galerkin method applied to the classical linear elastodynamic model Here, we consider the following system which is the strong formulation of the wave equation of elastodynamic (see in the Subsection 2.1) but we set ˙u=v, this is for an objective of numerical order. So, we have the following system                ρ∂v ∂t −∇·σ(∇u) = f, on Ω×]0, T[ ˙u=v, on Ω×]0, T[ u=gD, on ∂ΩD×]0, T[ n·σ(∇(u)) = gN, on ∂ΩN×]0, T[ u(x, 0) = u0(x), x ∈Ω v(x, 0) = v0(x), x ∈Ω. (3.2.1) In fact, the setting of ˙u=vallows us to discretize the system with the idea to have two different quantities of solutions which are uand v. 3.2.1 Function Spaces. Here, we define the spaces that we need in order to make a good discretization of our above system.  Displacement vector Since we have a non homogeneous boundary condition of Dirichlet, what we first define here is: Wu 0:= {u∈H1(Ω)d|u= 0 on ∂ΩD}. This is the set of function of H1but vanish on the Dirichlet boundary of Ω. Note.Here, the ”0” just means that we consider the homogeneous Dirichlet boundary condition, that is uvanishing on that Dirichlet boundary. Wu 0serves us as spatial test function spaces that we name in our work as displacement variation δu. The word spatial since we have not yet Section 3.2. Continous Galerkin method applied to the classical linear elastodynamic model Page 19 taken in consideration the temporal dimension, we will see down the final set taking in account the regularity of the test function in time. Now, we have to define our spatial space also within the ”u” solution displacement is going to live in space. To do that, we consider the space Wu={gD+Wu 0}. This is the spatial space of our solution, that we call here trial spatial space. We can clearly remark that in space, we have the test functions which are living in a different space from the trial functions due to the non homogeneous Dirichlet conditions. An importance of that is generally on the establishment of the variational formulation ( see in [4], Chapter 2 for more details ). Now, considering the L2regularity necessary in time of the solution, we define the sets: u∈ Hu=L2(I, W)and δu ∈ Hu 0=L2(I, Wu 0).  Velocity vector Concerning the velocity v, it worth mentioning that we have no spatial boundary conditions on v, and this means that we are not going as in the case of the displacement vector consider the space for test functions different from the one of the trial functions spatially. So, let consider the space Wv=L2(Ω)d, this is the spatial space for the test function or velocity variations denoted here by δv and also for the trial functions vand as done in the previous part and considering the regularity in time of the velocity, we define the space: Hv:= L2(I, Wv). After defining this set we can now claim our final goal in the next parts which will be to find a solution of our weak formlation that we will give down in the following specific sets and solutions, u∈ Hu,v∈ Hvand ∂v ∂t ∈L2(I, (Wu)∗). We have (Wu)∗is the dual space associated to Wu, and the latter partial derivative is taken in this set. 3.2.2 Weak Formulation. We define the weak formulation and give a proof of it, this is an expression deriving from the strong formulation above and expressed in the function spaces that we have defined just above. The definition of this weak formulation is relevant for the discretization scheme, the key of our interest for the following parts. 3.2.3 Proposition. The weak formulation associated to the strong formulation is given by finding u∈ Hu,v∈ Hvsuch that ZIZΩ ρ∂tv·δudV dt +ZIZΩ σ(∇u) : ∇δudV dt +ZIZΩ (∂tu−v)·δvdV dt =ZIZΩ f·δudV dt +ZIZ∂ΩN gN·δudAdt, with ∀δu ∈ Hu 0, δv ∈ Hvand initial conditions u(x, 0) = u0and v(x, 0) = v0in Ω,f:Ω−→Rd, gN:∂ΩN−→Rd. Section 3.2. Continous Galerkin method applied to the classical linear elastodynamic model Page 20 Proof. We consider the system in equation 3.2.1 above which is the strong formulation of the classical system of elastodynamic wave equation. By multiplying by δu, the first line and by δv the equation ˙u=v, the test functions,and integrating simultaneously over Ω×I we get: ∀δu ∈ Hu 0, δv ∈ Hv: ZIZΩ ρ∂tv·δudV dt −ZIZΩ (∇ · σ)·δudV dt =ZIZΩ f·δudV dt, (3.2.2) ZIZΩ (∂tu−v)·δvdV dt = 0. But lets remark that by integrating by parts over space the integral of (∇ · σ)·δu, we get: ZΩ (∇ · σ)·δudV =−ZΩ σ:∇δudV dt +Z∂ΩN (σ·n)·δudV. (3.2.3) Normally, we should also have a term over the Dirichlet boundary but since the test function δu(δu∈Hu 0) vanishes over the Dirichlet boundary(∂ΩD), we have the integral over the Dirichlet boundary which does not appear on the above expression. So, by replacing the equation 3.2.3 in the equation 3.2.2, we get: ZIZΩ ρ∂tv·δudV dt +ZIZΩ σ:∇δudV dt =ZIZ∂ΩN gN·δudAdt +ZIZΩ f·δudV dt, ZIZΩ (∂tu−v)·δvdV dt = 0. (3.2.4) Finally, by addition of the two latest equations, we get the weak formulation: ZIZΩ ρ∂tv·δudV dt +ZIZΩ σ(∇u) : ∇δudV dt +ZIZΩ (∂tu−v)·δvdV dt =ZIZΩ f·δudV dt +ZIZ∂ΩN gN·δudAdt, with u(x, 0) = u0(x)and v(x, 0) = v0(x). 3.2.4 Continuous Galerkin Discretization. Our goal here is to describe mathematically the continuous Galerkin discretization of this classical case. To do that we are going to define a finite dimensional set in space in which we are going to project our problem in space and then use the continuous Galerkin finite element method in time to fully discretize our problem in space time. Also the opposite can be done starting by defining a temporal discrete space in which we can project our problem and then fully discretize in space but for implementation it is usually done by the first method to easily implement. More details can be found on the paper [3]. Section 3.2. Continous Galerkin method applied to the classical linear elastodynamic model Page 21  Spatial Finite Dimensional Element Space We consider a decomposition of the set ¯ Ωinto non overlapping elements Kwhich can be triangle elements in dimension 2 or tetrahedra elements in dimension 3 for example. We call by hkthe local mesh size given by the formula: hk=diam(K) = supx,y∈Kkx−ykand we define h:= maxK∈Th,where we set Th={K}K∈Ω, a decomposition into non-overlapping elements of ¯ Ω. Let us define Wh⊂Wube a finite dimensional space containing the polynomials of degree less than or equal to s where: Wh={uh∈C(¯ Ω)d|(uh)|K∈Ps(K)d,∀K∈Th}.(3.2.5) As we will see later it’s in this set that we will approximate the solution uh(t) and vh(t)∀t∈I in space, this discrete set contains a local basis of polynomial which can be Lagrange polynomial of degree at most s. In this method of Galerkin the displacement test functions are also taken in this space.  Temporal discretization We have our temporal interval which is I, lets decompose our interval Iinto Mnon overlapping subintervals such that I=SM m=1, where: Im= [tm−1, tm], km=tm−tm−1, k := maxm∈[|1,...,M|]km. –In each subinterval we use a polynomial of degree rfor the approximation of the solution which is assume continue in time. –But for the approximation of test functions we assume to use polynomial of degree r−1since the test functions in the continuous Galerkin method are assumed globally discontinuous in time. Indeed due to the globally continuity of the trial functions we have the trial functions which by the obligation to linked subintervals by continuity is going to fixed a value at tmfor each subinterval which is going to make us lost one degree of freedom by that condition in tmso to have a system well-posed we just need test functions of order r−1in each subinterval. More details can be found in [3]. Lets recall our weak formulation by now adding the initial conditions imposed to the system in the weak formulation but we should pay attention on how adding them. Lets take in consideration the weak formulation which is equivalent to the one in Proposition (3.2.3): M X m=1 ZImZΩ ρ∂tv·δudV dt + M X m=1 ZImZΩ σ:∇δudV dt +ZΩ ρv(x, 0) ·δvdV =ZΩ ρv0·δvdV + M X m=1 ZImZ∂ΩN gN·δudAdt + M X m=1 ZImZΩ f·δudV dt , M X m=1 ZImZΩ ( ˙u−v)·δvdV dt +ZΩ ρu(x, 0) ·δudV = + ZΩ ρu0·δudV. Section 3.3. Continous Galerkin method applied to our unusual model Page 22 Note.We can remark like we said up the additions of the initials conditions, specifically the initial displacement has been added in two equations from 3.2.4 but here discretized, just because they represent respectively initial conditions on ufor the first degree differential equation in uand the initial condition on the velocity for the same reason on the first equation. –Temporal Finite Element Space :We are going to define some important spaces to discretize in time our weak formulation given above. * First of all for the trial functions, we define the set as Wc h,k := {u∈C0(¯ I, Wh)|u|Im∈Pr(Im, Wh),∀m= 1,2, ..., M}.(3.2.6) This is the set where leave the trial functions which are the functions solutions in space time. The parameter cis just to mean globally continuous functions. * We also define the set of test functions by: Wd h,k := {u∈L2(¯ I, Wh)|u|Im∈Pr−1(Im, Wh),∀m= 1,2, ..., M}.(3.2.7) Here, we need the L2regularity of the test function looking at our variational formulation. This is where live the discontinuous (the reason of the ”d”) function tests (discontinous in tm). 3.2.5 Remark. We can remark here our use of the set Whwhich is not unusual, this traduce the space-time discetization in the continuous Galerkin method. –Fully Discrete Variational Formulation Finally, we can define the fully discrete variational formulation by the space time continuous Galerkin method as ∀(δu, δv)∈Wd h,k ×Wd h,k. Then, we seek to find the couple solution (u, v)∈Wc h,k ×Wc h,k such that M X m=1 ZImZΩ ρ∂tv·δudV dt + M X m=1 ZImZΩ σ:∇δudV dt +ZΩ ρv(x, 0) ·δvdV =ZΩ ρv0·δvdV + M X m=1 ZImZ∂ΩN gN·δudAdt + M X m=1 ZImZΩ f·δudV dt, M X m=1 ZImZΩ (∂tu−v)·δvdV dt +ZΩ ρu(x, 0) ·δudV =ZΩ ρu0·δudV. (3.2.8) 3.3 Continous Galerkin method applied to our unusual model In this section, we first define the function spaces to perform a good discretization. 3.3.1 Function spaces. Similar to the previous section, we define the function spaces necessary to well define our weak formulation. We recall: Wu 0:= {u∈H1(Ω)d|u= 0 on ∂ΩD}and Wu={gD+Wu 0}. Section 4.2. Simulation results Page 29 4.2.2 Simulations related to the second datasets.  Classical problem: Here we consider the case where we recall our body force f is null, and our approximating polynomials for both solution uand vare of degree 1, so we get these tables resuming the L2error approximation following the different refinement cases which are given in Table 4.5. Table 4.5: L2errors for the classical problem with linear approximation for uand linear for v Space-time refinement (Nx, Nt)L2error (50,50) 1.6537×10−1 (100,100) 3.8460×10−2 (200,200) 9.6853×10−3 (400,400) 2.4833×10−3 Space refinement (Nt= 50) NxL2error 100 1.5552×10−1 200 1.5530×10−1 400 1.5529×10−1 800 1.5530×10−1 Time refinement (Nx= 50) NtL2error 100 3.7727×10−2 200 6.0728×10−2 400 5.7996×10+0 800 5.5347×10+0 1600 5.7248×10+0 3200 5.2219×10+0 6400 3.3072×10+0 We consider now a quadratic approximation for the solution uand a linear approximation for the solution v, and conduct numerical results with the same dataset 2 the results from the simulations are given in Table 4.6. Table 4.6: L2errors for the classical problem with quadratic approximation for the discrete solution u and a linear approximation for the discrete solution v Space-time refinement (Nx, Nt)L2error (50,50) 1.4463×10+1 (100,100) 1.3816×10+1 (200,200) 3.0965×10+1 (400,400) 7.6830×10+0 Space refinement (Nt= 50) NxL2error 100 3.2670×10+1 200 7.2167×10+1 400 1.5478×10+2 800 3.8398×10+2 1600 9.9271×10+2 3200 1.9212×10+3 Time refinement (Nx= 50) NtL2error 100 1.0137×10+1 200 3.5467×10+0 400 1.1450×10+0 800 6.3839×10+0 1600 3.1088×10−1 3200 1.3992×10−1  Unusual problem: Now, we consider our unusual problem case and we assume that the discrete solution of uand vare approximated by polynomials of degree 1 and we have the following results from the conducted simulations for the L2error in Table 4.7. Section 4.3. Analysis and insights Page 30 Table 4.7: L2errors for the unusual problem with linear approximation for uand linear for v Space-time refinement (Nx, Nt)L2error (50,50) 8.6635×10−2 (100,100) 2.0981×10−2 (200,200) 5.2072×10−3 (400,400) 1.2965×10−3 Space refinement (Nt= 50) NxL2error 100 8.5961×10−2 200 8.5512×10−2 400 8.5476×10−2 800 8.5469×10−2 1600 8.5468×10−2 3200 8.5467×10−2 6400 8.5467×10−2 Time refinement (Nx= 50) NtL2error 100 7.4393×10−2 200 4.2088×10−2 400 6.4414×10−3 800 2.5798×10−3 1600 2.2513×10−3 3200 2.2124×10−3 6400 2.2030×10−3 We still consider our unusual problem case and we assume that the solution of uis approximated by quadratic polynomials and the solution of vis approximated by linear polynomials, by conducting simulation we have the following results in Table 4.8. Table 4.8: L2errors for the unusual problem with quadratic approximation for uand linear for v Space-time refinement (Nx, Nt)L2error (50,50) 8.0690×10−1 (100,100) 1.3994×10+0 (200,200) 2.2242×10+0 (400,400) 3.1573×10+0 Space refinement (Nt= 50) NxL2error 100 5.7071×10−1 200 3.1877×10−1 400 1.6459×10−1 800 8.3764×10−2 1600 4.3623×10−2 3200 2.4871×10−2 6400 1.7236×10−2 Time refinement (Nx= 50) NtL2error 100 1.1615×10+0 200 9.1392×10−1 400 5.4738×10−1 800 2.9286×10−1 1600 1.4444×10−1 3200 6.6250×10−2 6400 2.7781×10−2 4.3 Analysis and insights In order to compare the atypical wave equation with an extra final time velocity condition to the classical wave equation with starting displacement and velocity conditions, we focus on the numerical simulation results shown in Tables 4.1–4.8. The L2error behavior, the influence of the final velocity condition, polynomial degree effects, dataset differences, and the relative contribution of time versus space refinement are the main topics of the investigation. With linear (degree 1) and quadratic (degree 2) Lagrange elements for the displacement uand velocity vin a mixed function space, the simulations were carried out using the continuous Galerkin finite element (cG-FEM) implemented in FEniCS with our two datasets defined above. 4.3.1 Error Behavior for Different Datasets. Here, we give some analysis on the error behavior depending of the dataset. First of all, we consider the dataset 1 and discussed the error of the classical problem and unusual problem with linear approximation below. The classical case with linear elements is presented in Table 4.1. This exhibits significant error reduction under space-time refinement, with L2errors decreasing from 3.7018 ×10−3(Nx=Nt= 50) to 6.6875 ×10−5(Nx=Nt= 400). Speaking about the convergence order, we compute the ratio of errors of consecutive mesh size usually from a mesh size and its double, if in average it is around Section 4.3. Analysis and insights Page 31 3.5-5.5 (e.g., here we have 3.7018×10−3 7.4361×10−4≈4.98) which is the error reduction’s factor, it is suggesting a quadratic convergence so a convergence of order 2. We recall that for defining the order of convergence p of a numerical scheme in general, we can take two errors from successive mesh size N= 50, E1 and N2= 100, E2and we have: p ≈log(E1 E2) log(N2 N1)for example in our case above p≈2,3that is why we said to be close to the quadratic convergence. However, space-only refinement shows minimal error reduction (from 2.0501 ×10−3) to 1.9110 ×10−3, suggesting that accuracy is limited by the temporal discretization. Moreover, time only refinement shows erratic behavior with error increasing significantly from Nt≥400,2.8768 ×10−1, likely due to the lack of spatial resolution. The Unusual case with linear elements is given in Table 4.3. The practically constant errors (around 7.7×10−2) obtained via space time-refinement point to a lack of convergence, which might be caused by the global influence of the condition of ending time velocity v(x, 4) = cos(4)x(1 −x). Temporal resolution is crucial for capturing the final time condition, as evidenced by the slight increase in errors from space-only refinement to 7.9225 ×10−2and the significant reduction from 7.1580 ×10−2to 8.1999 ×10−3from time-only refinement. Secondly, we considered the dataset 2 and discussed the approximation error on the classical case and unusual case with linear approximation polynomial. This finding are presented as in the following. The classical case with linear elements is given in Table 4.5. It presents a strong convergence under space-time refinement here, with errors dropping from 1.6537 ×10−1to 2.4833 ×10−3. This results in a error reduction’s factor of roughly 4 (e.g., 1.6537×10−1 3.8460×10−2≈4.30, which again suggests a quadratic convergence as properties. Time-only refinement leads to increasing errors for Nt≥400, reaching 5.7996 ×100indicating a certain numerical instability, whereas space-only refinement shows stagnant errors(around 1.55 ×10−1). The unusual case is presented in Table 4.7. This shows better convergence over space-time refinement (from 8.6635 ×10−2to 1.2965 ×10−3,factor of the error reduction ≈4) and time-only refinement reduces error to 2.2030 ×10−3whereas space-only refinement produces nearly constant errors(around 8.54 ×10−2). By comparing simulation results over polynomial degrees, we notice that:  Quadratic element for u(with linear for v) in the classical case (Tables 4.2 &4.6) generally produce higher order errors than the linear case, especially in dataset 2 (e.g., 1.4463 ×101,Table 4.6vs. 1.6537 ×10−1with approximation of u and v,Table 4.5at Nx=Nt= 50).  For the unusual case (Tables 4.4 &4.8), quadratic element for uleads to significantly larger errors (e.g., 5.8788 ×101,Table 4.8vs. 7.7138 ×10−2,Table 4.7for dataset 2, Nx=Nt= 400), suggesting that the final time condition amplifies the numerical challenges with high-order elements. Trends: The classical case consistently shows better convergence over space-time refinement, while the unusual case often exhibits higher errors or slower convergence, particularly (higher) in dataset 1, likely due to the final time condition’s global influence and at this stage, we assume maybe the potential influence of the term f. Time-only refinement is more effective in the unusual case, especially with linear elements, suggesting that temporal accuracy is critical for handling the final condition. 4.3.2 Impact of Final Velocity Condition. Our unusual wave equation incorporates a final time velocity condition v(x, T) = v∗ T, which imposes a constraint at the end time interval (T= 4 for both Section 4.3. Analysis and insights Page 32 datasets ). This condition significantly affect error magnitudes and convergence rates. In dataset 1, the unusual case with linear element (Table 4.3) shows nearly constant error under space-time refinement (≈7.7×10−2) and space-only refinement. Unlike the classical case’s steady decrease (Table 4.1, 3.7018 ×10−3to 6.6875 ×10−5). This suggest that the final time condition introduces a global constraint that limits the spatial convergence, as the solution must satisfy v(x, 4) = cos(4)x(1 −x), affecting the entire space-time cylinder. Time-only refinement (Table 4.3) reduces errors significantly (to 8.1999 ×10−3), indicating that final temporal meshes helps to capture the final condition accurately. For the dataset 2, the unusual case (Table 4.7) shows better convergence over space-time refinement with (rate ≈4) compared to the dataset 1, possibly due to the zero body force (f= 0) which reduces numerical complexity. Moreover, errors in the unusual case become lower than those from the classical case whatever the polynomial approximation order of u(quadratic or linear) and for all types of refinement (e.g.,space-time refinement 1.2965 ×10−3,Table 4.7vs. 2.4833 ×10−3at Nx=Nt= 400,Table 4.5), suggesting again that the lack of the body force f is an important factor for allowing the unusual case to perform or even outperform the classical case in accuracy for this dataset. The final condition introduces numerical challenges in the space-time cG-FEM, as the weak formulation includes boundary terms at t=T(see Proposition 3.3.3), requiring the solution to align globally across time. This can lead to error accumulation, especially with coarser meshes in time, as the solver balances initial and final conditions of time simultaneously. However in dataset 2, the simply dynamics enable the final time condition to act as an additional constraint that refines the solution, with sufficient mesh refinement. 4.3.3 Polynomial Degree Effects. Using quadratic elements for u(with linear v) generally increases errors compare to the linear element for both cases, contrary to expectations, as higher-order element typically improve accuracy. For the classical case in dataset 1 (Table 4.2), space-time refinement yields higher errors (e.g., 1.1152 ×10−1at Nx=Nt= 50) than linear elements (Table 4.1,3.7018 ×10−3 at Nx=Nt= 50). In dataset 2 (Table 4.6), errors are even larger (e.g., 1.4463 ×101). For the unusual case, quadratic elements (Tables 4.4 &4.8) produce significantly larger errors, especially in dataset 1 (e.g., 5.8788 ×101,Table 4.4at Nx=Nt= 400). This suggests that quadratic elements for uintroduce numerical instability, possibly due to mismatched polynomial degrees between uand v so the linear approximation for vlimits overall accuracy, as the velocity field may dominates the error contributions in the wave equation. 4.3.4 Dataset Comparison.  Dataset 1 (f6= 0) features a non-zero body force f(x, t) = (2 −x(1 −x)) sin(t), a longer time interval (I= [0,4]), and an exact solution u(x, t) = sin(t)x(1 −x). This complexity leads to smaller errors in the classical case with linear elements (e.g., in space-time refinement Table 4.1, 6.6875 ×10−5, Nx=Nt= 400) but poor convergence in the unusual case (e.g., in space-time refinement Table 4.3,≈7.7×10−2, Nx=Nt= 400). So the non-zero body force introduces additional dynamics, increasing sensitivity to mesh refinement and polynomial degree.  Dataset 2 (f= 0) has a simpler sinusoidal solution (u= sin(πt) sin(πx)) and a longer time interval (I= [0,4]), leading to better convergence in both cases. The classical case (Table 4.5) achieves errors as low as 2.4833 ×10−3, and the unusual case (Table 4.7) reaches 1.2965 ×10−3. The absence of the body force simplifies the dynamics, reducing numerical errors and allowing the final time condition to be capture more easily in the unusual case. Section 4.4. Discussion Page 33 The non zero body force in dataset 1 may increases error magnitudes, particularly in the unusual case where the final time condition and the natural dynamic forcing interact to limit convergence. The dataset 2’s simpler setup highlight the numerical method’s ability to handle the final time condition more effectively when external forces are absent. 4.3.5 Time vs. Space Refinement. The results indicate that time refinement is more critical than space refinement for both cases, especially for the unusual case. In dataset 1, classical time-only refinement (Table 4.1) shows erratic behavior, with errors increasing for Nt≥400, suggesting that fixed spatial resolution (Nx= 50) limits accuracy. In contrast, for the unusual case (Table 4.3) benefits significantly from time refinement, reducing errors from 7.1580 ×10−2to 8.1999 ×10−3. In dataset 2, time refinement in the unusual case (Table 4.7) is highly effective(the error drops to 2.2030 ×10−3), while the classical case (Table 4.5) shows instability. Space-only refinement consistently yields stagnant or slightly increasing errors in both cases and datasets, indicating that temporal discretization dominates error contributions which may due to the wave equation’s time-dependent nature and the final time condition’s global effect in the unusual case. Balanced space-time refinement is necessary for optimal convergence, but temporal accuracy is paramount for capturing the dynamics accurately. 4.4 Discussion 4.4.1 Key Findings. The previous part on the analysis of the numerical simulation in subsection 4.3 underscore the profound impact of the source term f(body force) and the critical role of the time refinement in the error analysis of the classical and unusual wave equations using the continuous Galerkin element method(cG-FEM). In dataset 1, with a non-zero source term f, the classical wave equation equation exhibits robust convergence, while the unusual wave equation, constrained by the final time velocity condition v(x, 4) = cos(4)x(1 −x), shows stagnant error (Table 4.3), indicating that the complex dynamics introduced by fhinder convergence. In contrast, dataset 2, with f= 0, allows the unusual case to achieve superior accuracy, reducing errors to 1.2965×10−3compare to 2.4833×10−3 for the classical case(Table 4.5 &4.7), suggesting that simpler dynamics enhance the final condition’s stabilizing effect. Also time-only refinement is particularly effective for the unusual case, slashing errors from 7.1580 ×10−2to 8.199 ×10−3in dataset 1(Table 4.3) and to 2.2030 ×10−3in dataset 2(Table 4.7), revealing that high temporal resolution is essential to capture the global influence of the final velocity condition. This provide an intuition that fine temporal meshes are crucial for mitigating error accumulation in non standard temporal constraints. Surprisingly one another thing is that, higher order approximations, such as quadratic polynomials for displacement u(with linear v), yield significantly larger errors,reaching 5.8788×101for the unusual case (Table 4.4) possibly due to the a certain cG scheme’s sensitivity to mismatched polynomial degrees or its inherent structure. 4.4.2 Limitations. This study faces several limitations that impact its generalizability. Foremost is the lack of a theoretical a priori error estimate for the unusual wave equation with final time velocity conditions, which complicates predicting convergence behavior or quantifying the initial condition’s effect, unlike the classical wave equation, where established analyses exist [7]. A theoretical prove of that error for our unusual case could be inspired from that prove in [7], by a rigourous analysis. Additionally, Section 4.4. Discussion Page 34 the numerical experiments reveal significant issues with high order polynomial approximation: quadratic approximations for displacement u(with linear for v) produce large errors(e.g.,5.8788 ×101,Table 4.4), and also combinations like quadratic polynomials for both uand v, or quadratic vwith linear u, are not even compiling when implementation in python, likely due to numerical instability from mismatched polynomial degrees or the cG scheme’s formulation. Which may not adequately handle higher-order terms in time-dependent problems. The cG scheme itself then may be a limitation, as its continuous temporal discretization may exacerbate errors when higher-order elements are used, particularly for the unusual case’s non-standard boundary conditions. Also it worth mentioning to say that computational constraints further restrict the analysis, with mesh refinement capped at Nx=Nt= 400 for space-time and Nt= 6400, especially for the unusual case where temporal accuracy is critical. These limitations underscore the need for both theoretical advancements and improved numerical strategies. 4.4.3 Future Work. Future research should prioritize developing a priori error estimates for our unusual wave equation, adapting frameworks from the classical case (see [7]) to quantify the final time velocity condition’s impact and guide numerical improvements. To address the poor performance of higher order approximation and testing balanced polynomial degrees as cubic or even quartic elements for both u and v, which leads to numerical instability, the use of alternative schemes like discontinuous Galerkin methods, which may better handle the cG scheme’s limitations with non standard conditions as done in ([9]) where the advantage of using discontinuous Galerkin method in space-time has been shown for second order hyperbolic problems. Implementing adaptive mesh refinement, which dynamically adjust spatial and temporal meshes based on error indicator as done in [3] for a Galerkin discretization of the classical linear wave equation, could also enhance efficiency and accuracy, particularly for the unusual case. Conclusion This thesis provides important insights into the numerical behavior of the classical and unusual wave equations by demonstrating the effectiveness and difficulties of using the continuous Galerkin method to solve them. While the atypical wave equation, which was limited by a final time condition, demonstrated the crucial importance of time-only refinement and the influence of the source term on error stagnation when non-zero, the classical wave equation showed strong convergence. However, the unexpectedly low displacement performance of quadratic approximations highlights numerical instability issues that may be related to the design of the cG scheme. These discoveries enhance our knowledge of space-time finite element techniques and demonstrate their potential for use in applications related to mechanics and thermodynamics. 35 Acknowledgements I wish to express my profound gratitude to all who have contributed to the successful completion of my master’s thesis. First and foremost, I am deeply thankful to the Almighty for granting me the strength and determination to undertake and complete this academic endeavor. My heartfelt appreciation goes to my supervisor, Prof. Dr. Thomas Wick, for his exceptional mentorship and steadfast commitment to my research. His insightful guidance, timely feedback were invaluable, pushing me to refine my ideas and achieve my goals. I extend my sincere thanks to Prof. Mama Foupouagnigni, Dr. Daniel Tcheutia, and Dr. Ignace Aristide Minlend for their valuable advice and encouragement during my research journey. Their expertise and supportive insights helped me stay motivated and navigate the complexities of my work. I am immensely grateful to my mentor, Dr. Kewani Welay Brhane, for his unwavering support and encouragement throughout my time at this institution. His wisdom and belief in my potential provided a steady anchor, inspiring me to tackle the challenges of this journey with confidence. I owe an immense debt of gratitude to my parents, BILONG BI POHA Louis Dieudonne and NGWE Anne Edwige , who have shaped me into the person I am today. From a young age, they instilled in me an unyielding work ethic, teaching me that hard work is the foundation of success. Their lessons in perseverance, love, and integrity have been my compass, guiding me through every endeavor. Their constant encouragement, not only in academics but in all aspects of life, has given me the courage to pursue my ambitions relentlessly, and for that, I am forever grateful. I also extend my sincere thanks to my classmates, tutors, and friends for their camaraderie and support. Their shared moments of motivation and collaboration made this thesis journey meaningful and memorable. I am particularly grateful to the administrative staff of AIMS Cameroon for their constant availability and assistance whenever we needed them. To all who have played a part in this milestone, I offer my deepest thanks for helping me bring this work to fruition. 36 References [1] Martin Alnæs, Jan Blechta, Johan Hake, August Johansson, Benjamin Kehlet, Anders Logg, Chris Richardson, Johannes Ring, Marie E Rognes, and Garth N Wells. The fenics project version 1.5. Archive of numerical software, 3(100), 2015. [2] Holm Altenbach. Kontinuumsmechanik. Springer, 2012. [3] Wolfgang Bangerth, Michael Geiger, and Rolf Rannacher. Adaptive galerkin finite element methods for the wave equation. Computational Methods in Applied Mathematics, 10(1):3–48, 2010. [4] Susanne C Brenner. The mathematical theory of finite element methods. Springer, 2008. [5] Haim Brezis and Haim Br´ezis. Functional analysis, Sobolev spaces and partial differential equations, volume 2. Springer, 2011. [6] Philippe G Ciarlet. Linear and nonlinear functional analysis with applications. SIAM, 2025. [7] Donald French and Todd Peterson. A continuous space-time finite element method for the wave equation. Mathematics of Computation, 65(214):491–506, 1996. [8] William Rowan Hamilton. On a general method in dynamics. Richard Taylor United Kindom, 1834. [9] Thomas JR Hughes and Gregory M Hulbert. Space-time finite element methods for elastodynamics: formulations and error estimates. Computer methods in applied mechanics and engineering, 66(3):339–363, 1988. [10] Philipp Junker and Daniel Balzani. An extended hamilton principle as unifying theory for coupled problems and dissipative microstructure evolution. Continuum Mechanics and Thermodynamics, 33(4):1931–1956, 2021. [11] Philipp Junker and Thomas Wick. Space-time variational material modeling: a new paradigm demonstrated for thermo-mechanically coupled wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage. Computational Mechanics, 73(2):365–402, 2024. [12] Julian Roth, Philipp Junker, and Thomas Wick. Space-time variational material modeling: Numerical simulations for the wave equation with velocity initial and final time conditions. 37