Higher order unfitted isoparametric space-time FEM on moving domains

Transcription

1 Higer order unfitted isoparametric space-time FEM on moving domains Master s tesis by: Janosc Preuß Supervisor: Jun.-Prof. Dr. Cristop Lerenfeld Second Assessor: Prof. Dr. Gert Lube prepared at te Institute for Numerical and Applied Matematics University of Göttingen Version as of (including minor corrections)

2 Hiermit erkläre ic, dass ic die vorliegende Arbeit selbstständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet abe. Göttingen, den 31. Januar 2018

3 Acknowledgments First of all, I would like to express my gratitude to my supervisor Jun.-Prof. Dr. Cristop Lerenfeld for proposing suc an exciting topic for my tesis and guiding my work trougout its preparation wit great commitment. I feel grateful for aving a supervisor wo is willing to discuss open problems and provide insigtful answers to my questions at nearly any time of te day. His entusiasm and support ave been immensely motivating trougout te last year. I would also like to tank Prof. Dr. Gert Lube for kindling my interest in numerics of partial differential equations. My Bacelor s tesis, tat I prepared under is supervision, introduced me to advanced finite element metods and paved te way for te tesis at and. I also appreciate tat e examines tis tesis as a second assessor. Te numerical experiments in tis tesis ave been performed wit te finite element software Netgen/NGSolve and its Add-On ngsxfem for unfitted discretizations. I would like to take te oppurtunity to tank all people wo ave contributed to developing tis software. In particular, I would like to point out Fabian Heimann wo is working on extensions of te numerical integration in ngsxfem. His work forms te backbone for te numerical realization of te metod presented in tis tesis. At last, I would like to sincerely tank my family for teir support and encouragement during my studies.

4 Contents 1 Introduction Motivation Outline of te tesis Space-time discretization for a moving domain problem assuming exact geometry andling Model problem Space-time discretization Error analysis of a space-time discretization for a moving domain problem assuming exact geometry andling Notation and assumptions Near best-approximation in a discrete norm Norms and inverse estimates Stability Continuity Consistency Céa-like result Interpolation in space-time Interpolation in tensor-product space-time spaces Interpolation in unfitted space-time finite element spaces A priori error estimate in discrete norm Isoparametric space-time discretization for a moving domain problem Isoparametric (unfitted) FEM Space-Time mes deformation Construction Discontinuity of mes deformation between time slabs Test problem: Circle moving troug mes Isoparametric space-time discretization Implementational aspects Space-time finite element spaces Quadrature on space-time level set domains Numerical experiments for a moving domain problem Moving circle Description of test case Convergence tables

5 6.1.3 Discussion of results Moving and deforming ellipse Description of test case Discussion of results Isoparametric unfitted space-time FEM for a two-pase interface problem Introduction Derivation of te metod Numerical experiments for a two-pase interface problem Moving (curved) plane Moving circle Conclusion Summary Open problems and outlook

6 Capter 1 Introduction 1.1 Motivation Solving partial differential equations (PDE) on evolving geometries is a callenging and useful task. Te geometry tat as to be treated in applications is often quite complicated. Tis is for example te case in te simulation of two-pase flows. Here te interface tat separates te two fluids may undergo large deformations. Even topology canges are possible wen droplets emerge. Te efficient and accurate simulation of suc penomena requires te development of novel numerical tecniques. Te standard finite element metod (FEM) employs a mes tat is fitted to te geometry. Keeping te mes conforming to a moving and deforming geometry may require substantial efforts. Tus, it is interesting to consider unfitted finite element metods were te mes is independent of te geometry on wic te PDE as to be solved. Since remesing procedures are avoided, tis approac seems to be a promising basis for constructing efficient numerical tecniques. Using an unfitted metod entails new callenges. Often one cooses to model te geometry by means of a level set function. Tis provides a igly accurate, yet implicit description of te domains on wic te integrals arising in te variational formulation of te PDE ave to be calculated. Since te mes is not fitted to te geometry, an implementation of suc metods tus requires to evaluate integrals on cut elements. Te cut is described only implicitly as te zero set of te level set function. If linear finite elements are used, ten it suffices to compute tese integrals wit second order accuracy in order to preserve te accuracy of te wole metod. Since tis task can be solved robustly by establised metods, most researc so far as concentrated on linear finite elements. However, te extension to iger order elements is not straigtforward and requires new ideas. In [Le16] and [LR17], a new approac for ig order unfitted finite element metods as been proposed and analyzed for stationary problems. It is based on a parametric mapping of te underlying mes. On simplicial meses, te image of a piecewise planar representation of te geometry under tis mapping yields a ig order accurate description of te geometry. Te resulting isoparametric unfitted finite element metod as been sown to allow for error bounds of optimal order. In tis tesis, we start to extend tis approac to evolving geometries. 1

7 1.2 Outline of te tesis As a model problem, we consider a convection diffusion equation on a moving domain. Te first six capters deal wit te development of a iger order isoparametric FEM for tis problem. Te main part of te tesis consists of an a priori error analysis for te derived metod (under te assumption of exact geometry andling) and numerical experiments. In te end, we also outline ow te metod can be applied to mass transport problems in two-pase flows and present results of numerical experiments. Te outline of te tesis is as follows: In capter 2 a space-time discretization for te moving domain problem is derived. For te time stepping we use a discontinuous Galerkin metod [To97] applied in an unfitted setting. Here, te space-time domain is partitioned into time slabs. We define a finite element space on tese time slabs and derive a variational formulation wic allows to solve te problem time slab per time slab. In tis capter it is assumed tat all te arising integrals can be calculated exactly. A description ow we acieve iger order geometrical accuracy in practice will be given in capter 4. In capter 3 an a priori error analysis for te metod from capter 2, wic assumes exact geometry andling, is carried out. We combine stability, consistency and continuity to derive a Céa-like result. Te formulation of te metod involves a finite element space on te time slabs. For proving an a priori error estimate a suitable interpolation operator into tese spaces is constructed. Making use of te interpolation results we arrive at an error estimate in a discrete norm wic is anisotropic in te time step and te spatial mes widt. Te previous two capters assumed an exact andling of te geometry, wic is unattainable in practice. Tus, capter 4 deals wit te extension of te isoparametric metod from [LR17] to te instationary case. We derive a space-time version of te parametric mapping from [LR17]. Ten it is sown ow te finite element spaces and te variational formulation on te time slabs need to be adapted in order to benefit from te iger order accurate geometry description provided by te parametric mapping. Tis results in an isoparametric unfitted space-time discretization. Some aspects regarding te implementation of te metod are discussed in capter 5. One of tem is ow te integration on space-time domains tat are implicitely described by a level set function is carried out. Te iger order accurate, isoparametric metod for stationary problems from [LR17] allows to reduce all te arising integrals to a reference configuration wic is described by a piecewise linear approximation of te level set function. Te situation for te extension to te space-time case is similar. In capter 6 te metod is tested for two different moving domains. Te results of te numerical experiments are compared to te a priori error estimate derived in capter 3. Te observed rates are better tan guaranteed by te derived error estimate. 2

8 Capter 7 illustrates ow te metod can be applied to mass transport problems in two-pase flows. Te conditions at te fluid interface are imposed by means of te Nitsce tecnique (see [Nit71],[HH02] and [RN09]). We derive te metod but do not provide an analysis. In capter 8 numerical experiments for two-pase interface problems are presented. As test cases a moving plane and a moving circle are considered. We conclude in capter 9 wit a summary of te tesis. Open problems will be discussed and we propose directions for furter researc. 3

9 Capter 2 Space-time discretization for a moving domain problem assuming exact geometry andling Tis capter starts by introducing te main model problem wic is considered in tis tesis. Ten we derive a variational formulation for tis problem. Time and space variables are treated similarly in tis formulation. Moreover, our metod is unfitted, i.e. te geometry does not fit to te mes. We assume an exact andling of te geometry in tis capter. Tis includes tat all te arising integrals can be calculated exactly. Tis assumption will be dropped in capter 4, were it is described ow we acieve iger order geometrical accuracy in practice. To tis end, we use an isoparametric finite element metod and an adapted variational formulation. Tese aspects will be introduced later in capter Model problem Let Ω(t) R d for d {1, 2, 3} be a time-dependent domain wose evolution is driven by a divergence-free convection field w (see Figure 2.1). Tis domain Ω(t) contains a quantity wose concentration is modelled by a scalar field u(x, t). We assume tat te concentration fulfills a convection-diffusion equation inside Ω(t). Moreover, we suppose tat te quantity contained inside te time-dependent domain does not flow out over te boundary, u n Ω(t) = 0, were n Ω(t) denotes te outer normal vector to Ω(t). Given te evolution of te domain Ω(t) and te initial concentration u 0 (x) at t = 0 te task is ten to compute te concentration u(x, t) for t > 0. So te problem is: t u u + w u = f in Ω(t), t [0, T ], w = 0 in Ω(t), t [0, T ], u n Ω = 0 on Ω(t), t [0, T ], u(, t = 0) = u 0 in Ω(t = 0). (2.1) Remark 1. In principle it is possible tat te quantity contained inside Ω(t) exerts a pressure on Ω(t). Furtermore, one could take te surface tension as a counteracting force into account. Tis is for example te case in an osmotic cell swelling problem, were te evolution of te cell is determined by tese two forces. In tis tesis we neglect 4

10 Ω(t 2 ) w Ω(t 2 ) Ω(t 1 ) Ω(t 1 ) Ω Figure 2.1: Te domain moving from t 1 to t 2 wit t 2 > t 1. tese complications and assume tat te velocity of te boundary in normal direction V n is solely determined by te given convection field, tat is V n = w n. 2.2 Space-time discretization Te problem will be treated by an unfitted space-time DG metod. In order to avoid d + 1-dimensional complexity for te arising algebraic systems, te space-time domain will be divided into so called time slabs. To tis end, let 0 < t 1 <... < t N 1 < t N be a partition of te time domain into time intervals I n = (t n 1, t n ]. For simplicity of presentation, te time intervals are of constant size t = t n t n 1. We introduce te time slab Q n := t In Ω(t) {t}. Te wole space-time domain is ten given by te union over te time slabs Q = N Q n. Te spatial domain at a fixed time Ω(t n ) will sometimes be abbreviated by Ω n = Ω(t n ). Furter, we define te space-time boundary Γ := t (0,T ] Ω(t) {t} and its restriction to te time slab Γ n := t I n Ω(t) {t}. Next we will derive a variational formulation on te time slabs Q n wic allows to solve te problem time slab per time slab. Tis leads to te variational structure of a time-stepping sceme. Tis requires a finite element space W n on te time slabs Q n. To define tis space some preparations are necessary. Let Ω be a larger, time independent, polygonal background domain tat contains Ω(t) for all times t. Te time slabs Q n are ten contained in Q n = Ω I n and Q is a subset of Q = N Qn. Let T n be a sape-regular triangulation of te background domain Ω. In tis tesis we will only work wit simplicial meses. Te index n indicates tat te triangulation is in principle allowed to cange between te time slabs. However, for ease of presentation, we restrict ere to te case of a fixed triangulation T n = T on every time slab n = 1,..., N. Te extension to non-matcing triangulations poses no major difficulties. Te elements T of te spatial mes and te time interval I n form te space-time prisms Q n T = T I n. Let V ks be a standard finite element space of order k s on te mes T, i.e. V ks := {v H 1 ( Ω) v T P ks (T ) T T }, 5

11 t n t n 1 t n t n 1 t n t n 1 t n t n 1 t t t t Γ Q n Γ E(Q n ) Γ I(Q n ) Γ ω(q n ) ω(q n ) Γ Γ Γ Γ x x x x Figure 2.2: Sketc of different domains on te time slab: Here we denote by ω(q n ) te space-time prisms tat are involved in te stabilization term, i.e. ω(q n ) = F F,n R ω F I n. were P ks (T ) denotes te space of polynomials up to degree k s on te simplex T. We define an extension operator E tat extends te domain Q n onto a domain wit a tensor product structure witin eac time slab E(Q n ) := {x T for some T T wit Q n T Q n }. (2.2) Let ten E(Q) = N (E(Q n )). We furter introduce an operator wic restricts te domain Q n onto a domain wit a tensor product structure witin eac time slab I(Q n ) := Q n \ E(( Q I n ) \ Q n ). Ten we introduce purely spatial counterparts I(Ω n ) suc tat I(Ω n ) I n = I(Q n ), E(Ω n ) suc tat E(Ω n ) I n = E(Q n ). Here E(Ω n ) migt extend into regions were elements are not touced by Ω n at time t n. Te different domains on te time slab are sketced in Figure 2.2 for te spatially onedimensional case. Figure 2.4 provides an illustration of E(Ω n ) for two spatial dimensions. Now we define te ansatz space W = {v : E(Q) R v E(Q n ) W n } wit W n defined as W n := {v : E(Q n ) R v(x, t) = k t m=0 t m φ m, φ m V ks (E(Ωn ))}. (2.3) Here V ks (E(Ωn )) denotes te restriction of V ks to te so called active mes E(Ω n ). Note tat W contains functions tat may be discontinuous between te time slabs. 6

12 t t n t n 1 Γ Q n 1 Q n Qn+1 u n 1 u n + u n 1 + Γ Figure 2.3: DG in time: Te functions in W may be discontinuous between te time slabs. Tis means tat te limit u n 1 coming from below migt not agree wit te limit u n 1 + coming above on Ω n 1. To derive te variational formulation, we multiply te PDE wit a function v W n and integrate over Q n. Due to omogeneous Neumann boundary conditions an integration by parts of te diffusion term ten leads to: ( t u + w u, v) Q n + ( u, v) Q n = (f, v) Q n. Tis equation, posed for all v W n, is not sufficient to specify te solution on te time slab since it does not involve an initial condition. Tere are different ways to include information from te previous time slab. One of tem is to recognize tat te time derivative acts as a convection term in te space-time domain Q n. In tis sense Ω n 1 {t n 1 } is te inflow boundary of Q n were inflow information as to ( be provided. ) Here one adds upwind stabilization to impose weak continuity in time : u n 1, v+ n 1. Ω n 1 Te terms v± n 1 are defined as te limits in time from above respectively below v± n 1 := lim s 0 v(, t n 1 ± s). An illustration is given in Figure 2.3. Te bracket denotes te jump over te time boundary u n 1 := u+ n 1 u n 1. Te term u n 1 is known from te previous time slab and can be sifted to te rigt and side. Tis leads to te variational formulation: Find u W n suc tat ( t u + w u, v) Q n + ( u, v) Q n + ( u+ n 1, v+ n 1 = (f, v) )Ω n 1 Q n + ( ) u n 1, v+ n 1 (2.4) Ω n 1 for all v W n olds. Te equation will be treated by an unfitted metod. Tat is, te mes is not fitted to Ω(t). A major advantage of tis approac is tat one can work wit a simple background mes tat does not need to be canged during te time evolution. In particular, possibly expensive remesing procedures are avoided. Unfortunately, te unfitted approac also gives rise to some difficulties. One of tem is ow to control te norm of te solution on te elements wic are cut by te boundary Ω(t). Certain inverse inequalities tat are known from te fitted case are not valid anymore in tis situation. In order to resolve tis difficulty we will add a stabilization term j n (u, v) to te variational formulation. It will allow us to regain control of te solution on te wole computational domain and essentially carry over te inverse inequalities from te fitted case. To tis end, it is necessary to introduce some furter notation. Let F = {F } be te set of spatial facets of T. Te relevant facets for te stabilization are ten given by F,n R := {F F : F = T 1 T 2, T 1 E(Ω n ) \ I(Ω n ), T 2 E(Ω n )}. 7

13 Te corresponding facet-patces are defined as ω F := T 1 T 2, F T i, i = 1, 2. (2.5) An illustration is given in Figure 2.2 for d = 1 and in Figure 2.4 for d = 2. Ten, we define te stabilization term j n (u, v) := t n t n 1 γ J ( t, ) F F,n R ω F 1 2 u ω F v ωf dx dt, (2.6) for discrete functions u, v W. Here, u ωf is used to denote te jump on te facet patc and is defined in te following way: Let ω F = T 1 T 2 and denote by u i te restriction of u to te element T i for i = 1, 2. Tese polynomials ave a canonical extension to te neigboring elements, e.g. te polynomial u 1 (x) can be evaluated at x T 2. Tis allows to define u ωf (x) := v 1 (x) v 2 (x) for x ω F. Te scaling factor γ J ( t, ) := γ J (1 ) + t will be motivated in capter 3. Its main function is to compensate for an anisotropic coice of te spatial mes widt and te time step. For t we ave γ J ( t, ) = O(1). Remark 2 (Relation of stabilization term to literature). Tere are (at least) two ways discussed in te literature on unfitted metods to gain control of te norm on cut elements. Te most popular one is a stabilization based on penalizing jumps of normal derivatives over element facets (e.g. [BH12]). Te relation between tis stabilization and te one used ere will be discussed in Remark 6. Anoter option is a local projection type stabilization wic goes back to [BB01] for te fitted case. In te context of unfitted metods it is for example applied in [Bur10] (section 4) and [BH14]. Tis stabilization consists of a sum of integrals over patces ω l (not necessarily associated to a facet) of te form 2 ω l (u P ωl u)v dx ω l for discrete functions u and v. Here, P ωl u is an L 2 - projection onto a polynomial on te patc ω l and ωl O() te diameter of te patc. Te stabilization used in tis tesis is similar because it also involves integrals over (facet) patces. However, we do not compare te functions to teir projection, but rater tere restrictions to te individual elements wic form te facet patc. Neverteless, an L 2 projection onto te facet patc will sow up later in te analysis wile bounding te approximation error of our stabilization (see Proposition 3.26). Te discrete variational problem on te wole space-time domain is obtained by summing up over te time slabs: Find u W suc tat for all v W tere olds wit B(u, v) := B(u, v) + J(u, v) = f(v) (2.7) ( t u + w u, v) Q n + N 1 ( + u n, v+) n + ( ) u 0 Ω +, v 0 n + 8 ( u, v) Q n Ω 0, (2.8)

14 and f(v) := J(u, v) := j(u, n v), (f, v) Q n + ( ) u 0, v+ 0 It is useful to define te abbreviations Ω 0. d(u, v) := ( t u + w u, v) Q n, N 1 ( b(u, v) := u n, v+) n + ( ) u 0 Ω +, v 0 n + a(u, v) := ( u, v) Q n. We also introduce variants wic we will use later d (u, v) := (u, t v w v) Q n, N 1 ( b (u, v) := u n, v n) + ( ) u N Ω, v N n Lemma 2.1 (Rewriting te bilinear form). For u, v W + H 1 (Q) tere olds d(u, v) + b(u, v) = d (u, v) + b (u, v) Ω 0, Ω N. and tus B(u, v) = d (u, v) + b (u, v) + a(u, v). (2.9) Proof. Te alternative representation of te bilinear form is derived by integration by parts of te space-time convection (, t ). For te time derivative one obtains ( t u, v) Q n = (u, t v) Q n + ( u n, v n )Ω ( ) u n 1 n +, v+ n 1 Ω n 1 were V n denotes te velocity of te boundary in normal direction. For te convection term one as (w u, v) Q n = (u, (wv)) Q n + t n t n t n 1 Ω(t) w nuv dsdt V n uv dsdt, = (u, w v) Q n + t n 1 Ω(t) t n w nuv dsdt, t n 1 Ω(t) 9

15 were it was used tat (wv) = v w + w v = w v since w is divergence free. Now one as tat te velocity of te boundary in normal direction coincides wit te convection field: w n V n = 0, cf. Remark 1. So tese two terms cancel and we obtain: ( t u + w u, v) Q n = (u, t v w v) Q n + ( u n, v n )Ω ( ) u n 1 n +, v+ n 1 Summation over n = 1,..., N leads to Writing d(u, v) = { ( u n, v n )Ω ( ) u n 1 n +, v+ n 1 } + d (u, v). Ω n 1 b(u, v) = = N 1 ( u n + u n, v n +) ( ) u n 1 +, v+ n 1 Ω n 1 Ω n + ( u 0 +, v 0 + N 1 ) Ω 0 ( u n, v n +) and adding tis to te expression for d(u, v) from above yields: b(u, v) + d(u, v) = N 1 ( u n, v n + v n ) = b (u, v) + d (u, v). Ω n + ( u N, v N ) Ω n Ω N + d (u, v) Ω n 1. Remark 3 (Mass conservation). Lemma 2.1 can be employed to sow tat te space-time DG metod is globally mass conserving. Using te caracterization of B from (2.9) and testing wit v = 1 in te variational formulation yields: Ω N Ω 0 Q u (, t N ) dx = u 0 dx + f dx. (2.10) Ω A local version of tis mass balance on eac time slab is obtained by testing wit a function v tat is constant on te time slab Q n and vanises everywere else: Ω n Q n u (, t n ) dx = u (, t n 1) dx + f dx. (2.11) 10

16 Ω(t n ) Ω(t n ) w Ω(t n 1 ) Ω(t n 1 ) Ω(t n ) w E(Ω n ) Ω(t n 1 ) Ω(t n ) w F F,n R ω F Ω(t n 1 ) Figure 2.4: Te topmost sketc sows te domain Ω(t) at times t = t n 1 and t = t n. In te middle, one can see te spatial domain E(Ω n ) wic forms te extended time slab E(Ω n ) I n = E(Q n ). Te facet patces involved in te stabilization term j n are sown in te lowermost sketc. 11

17 Capter 3 Error analysis of a space-time discretization for a moving domain problem assuming exact geometry andling Tis capter contains an error analysis for a modified version of te metod introduced in capter 2. It is assumed tat all te arising integrals can be calculated exactly. Te analysis proceeds by an approac tat is similar to te error analysis of DG scemes for linear transport equations. See for example te procedure for te linear model problem arising from DG metods described in section 1.3 of [DPE12]. First we define discrete norms. Ten we calculate te consistency error and sow boundedness and (discrete) stability. To tis end, we need certain inverse estimates tat are not automatically guaranteed for unfitted problems. At tis point te gost penalty stabilization comes into play. It basically allows to carry over te inverse estimates tat are known for te discretization of PDEs on fitted tensor product domains. Based on te derived properties of te bilinear form we prove a Céa-like result. In order to carry on, we need to bound te best approximation error of te solution in te employed space-time finite element spaces. Tis requires suitable interpolation operators. We construct tese and derive teir stability and approximation properties. Te approximation results are ten applied to prove an error bound for te metod tat is anisotropic in te time step and spatial mes widt as well as te spatial and temporal polynomial degrees. 3.1 Notation and assumptions Te analysis is based on assumptions A1-A5 wic will now be introduced. Te first assumption A1 is related to te use of an unfitted discretization. In order to accomodate for tis assumption, te set of facet patces {ω F F F,n R } tat are used for te stabilization needs to be expanded. We will first state te assumption and ten provide some explanation in a subsequent remark. A.1 Tere exists a mapping (between elements) B : E(Ω n ) I(Ω n ) suc tat: Te number of elements T E(Ω n ) tat map to a specific element T 0 I(Ω n ) can be bounded independently of and t, i.e. #(B 1 (T 0 )) C. 12

18 t n t n 1 t n t n 1 t t Γ E(Ω n ) \ I(Ω n ) I(Ω n ) Γ E(Ω n ) \ I(Ω n ) I(Ω n ) Γ Γ x x Figure 3.1: Illustration of assumption A.1: Te upper sketc sows a basic mes wile te lower one features a spatial refinement. Elements wit te same pattern are mapped to eac oter under B : E(Ω n ) \ I(Ω n ) I(Ω n ). Te facets for te stabilization term are marked in orange. For T E(Ω n )\I(Ω n ) let {T i } M i=0 be te set of elements tat need to be crossed in order to traverse from T M = T to T 0 = B(T ). Ten te facets {T i T j i, j = 1,..., M; i j} are contained in F,n R. Tat is, te stabilization encompasses all facets tat are passed wile walking from T to T 0. Te tickness of te layer of cut elements is bounded as ( #{T E(Ω n ) \ I(Ω n )} C B 1 + t ) wit C B independent of and t. (3.1) Remark 4. Tis assumption arises because we use an unfitted discretization. Here one needs to control te norm on cut elements T E(Ω n ) \ I(Ω n ). Combining tis assumption wit te employed stabilization will allow us to bound te norm on a cut element T against te norm on an uncut element B(T ) = T 0 I(Ω n ) plus stabilization terms. Te stabilization needs to be applied on all te facet patces tat need to be crossed in order to traverse from T to T 0. In te analysis, we need bounds on te number of elements #(B 1 (T 0 )) tat map to an element T 0 I(Ω n ) and on te tickness of te layer of cut elements #{T E(Ω n ) \ I(Ω n )}. Tese assumptions are illustrated in Figure 3.1: Te upper sketc sows a given cut situation on a coarse mes wile te lower sketc illustrates te beavior under mes refinement. Wit respect to assumption A.1 te following two observations can be made: On te coarse mes te boundary Γ on te left cuts troug te two elements wit a blue pattern. But on te finer mes te number of elements tat are cut by tis part of te boundary as increased to four. Hence, it is not plausible to impose a uniform bound on te number of elements tat are cut by te boundary. Requesting a bound of te form (3.1), wic imlies an increase wit srinking mes size, is a more realistic assumption. 13

19 On te oter and, we require a uniform bound on te number of elements tat map under B : E(Ω n ) \ I(Ω n ) I(Ω n ) to te same element T 0 I(Ω n ). Tis assumption is also illustrated in Figure 3.1. Here, te elements T E(Ω n ) \ I(Ω n ) and teir images B(T ) I(Ω n ) ave been drawn wit te same pattern. For example, te element wit te blue dots, wic is cut by Γ, on te coarse mes maps to te element wit te green dots in I(Ω n ). Bot on te coarse and te fine mes te Figure sows a one-to-one correspondence between cut elements and teir images in I(Ω n ). Tis illustrates tat te assumption #(B 1 (T 0 )) C for T 0 I(Ω n ) is not unrealistic. As mentioned above, tis assumption will be used to bound te norm on a cut element T E(Ω n )\I(Ω n ) against te norm on B(T ) I(Ω n ). In order to realize tis bound, we need to take into account all te facets tat ave to be crossed to walk from T to B(T ) and collect te corresponding stabilization terms. Hoewever, tese facets are not necessarily included in te original definition of te set F,n R given in Capter 2. For example, on te coarse mes, only te facet between te element wit te blue dots and te element wit te green orizontal lines would be included in F,n R, not te one between te element wit te green orizontal lines and its rigt neigbor. In practice, tis means tat we ave to expand te stabilization to a small band inside te domain wic includes approximately as many elements as are cut by te boundary in order to realize assumption A.1. Wit respect to te numerical experiments in Capter 6, we already mention tat tis enricment of F,n R as not been applied. Instead we stick to te original definition of F,n R given in Capter 2 for all numerical computations. Finally, ( we ) define te scaling factor γ J ( t, ) in equation (2.6) as γ J ( t, ) = γ J 1 + t wit γj > 0 independent of t and. Tis accounts for te factor obtained by te bound (3.1). A.2 We assume w is bounded. A.3 Tere exists a constant C G > 0 suc tat 2 t C G. A.4 Te time step is bounded by a constant t C o. A.5 Te domain Ω(t) for all times t and accordingly te space-time domain Q n are sufficiently smoot, so tat a linear continuous Soobolev extension operator exists. E n : H k (Q n ) H k (E(Q n )), k N, Remark 5. In view of te fact tat te spatial triangulation is required to be sufficiently fine to resolve te moving domain, te assumption A.3 appears to be rater mild. A reasonable coice for te constant in A.4 is C o = 1. In te analysis below, we labelled some constants, wic we would like to keep track of, according to te Lemma in wic tey originate. For example, C 3.3 is te constant corresponding to Lemma

20 T 1 v 1 = v T1 F T 2 v 2 = v T2 Figure 3.2: Sketc of te macro element T = T 1 T Near best-approximation in a discrete norm Norms and inverse estimates We introduce discrete norms. Te first norm is wit u 2 := u 2 := t( t u, t u) Q n + u 2 + N 1 ( u, u) Q n ( u n, u n ) Ω n + ( ) u 0 +, u ( ) u N Ω, u N 0 Let u 2 J := J(u, u) be te semi-norm induced by te gost penalty term. Ten we define te norm u 2 j := u 2 + u 2 J, wic offers additional control on te gost penalty term.tis norm is mainly used. For continuity we will also consider anoter discrete norm: wit u 2 := N ( ) 1 t u, u + u 2 + Q n N u 2 := ( u n, u ) n Ω N. ( u, u) Q n Also for tis norm tere is a corresponding version wic includes te gost penalty term: Ω n. u 2,j := u 2 + u 2 J. Te following Lemma sows te main mecanism beind te stabilizing effect of te gost penalty terms. It is a modification of te classical gost penalty result found in Lemma 5.1 of [MLLR14]. Lemma 3.1 (Gost penalty mecanism). Let T 1, T 2 T be two elements saring a common face F (illustrated in Figure 3.2). Moreover, assume tat v is a piecewise polynomial function of degree at most k s N, possibly discontinuous and defined relative to te macro element T = T 1 T 2. Denote by v i te restriction of v to T i for i = 1, 2. 15

21 Ten tere is a constant C > 0 wic only depends on te sape regularity of T and te polynomial order of v suc tat: v 2 T 1 C ( v ωf 2 T 1 + v 2 T 2 ) (3.2) olds true. Proof. Consider a point x T 1. We ave v 1 (x) = (v 1 v 2 )(x) + v 2 (x) = v ωf (x) + v 2 (x), were v 2 is te canonical extension of te polynomial defined on T 2 to T 1. Integrating over T 1 yields: v 1 2 T 1 2 ( v ωf 2 T 1 + v 2 2 T 1 ). Due to sape regularity te norms T1 and T2 are equivalent: v 2 T1 C v 2 T2. Tus, v 1 2 T 1 C ( v ωf 2 T 1 + v 2 2 T 2 ) wic completes te proof. Remark 6 (Comparison wit derivative-jump -based gost penalty). Let us investigate te proof of Lemma 5.1 from [MLLR14] wile keeping te notation from te proof of Lemma 3.1 above. Te autors proceed in tis proof by writing te polynomials v i (x) at a point x T 1 in terms of teir Taylor series around x F. Here, x F denotes te normal projection of x onto te plane defined by te face F. Tis yields v i (x) = Dα v i (x F ) ( x x F n) α, α! α k s were α = (α 1,..., α d ) is a multiindex and n α = n α 1 1 n α n α d d wit n = (x x F )/ x x F being te normal vector of te face F pointing towards T 1. Ten te two Taylor expansions are subtracted wic gives v 1 (x) = v 2 (x) + Dα v(x F ) ( x x F n) α, (3.3) α! α k s were D α v(x F ) = D α v 1 (x F ) D α v 2 (x F ) denotes te jump over te facet F. Now an integration over T 1 wit respect to x is performed, te expressions are squared and te Caucy-Scwarz inequality is used: v 1 2 T 1 C v 2 2 T ( D α v(x F (x)) n α ) 2 2 α dx. α k s T 1 After a cange of variables and an application of sape regularity te autors arrive at v 1 2 T 1 C v 2 2 T 1 + nv(y) j 2 2j+1 dy, j ks F 16

22 wit nv(y) j = D α v(y)n α. Using tat te norms v 2 2 T 1 and v 2 2 T 2 are equivalent by α =j sape regularity tis leads to te key result of te gost penalty mecanism ( ) v 2 T 1 C v 2 T 2 + 2j+1( nv(y), j nv(y) ) j. (3.4) F j k s If we perform a minor modification of tis argument by bringing v 2 (x) in (3.3) to te oter side we arrive at v 1 v 2 2 T 1 C j k s 2j+1( j nv(y), j nv(y) ) F. Ten ω F v 2 ω F dx = v 1 v 2 2 T 1 + v 2 v 1 2 T 2 2C j k s 2j+1( j nv(y), j nv(y) ) F. Tus, te stabilization used in tis tesis is bounded from above by te derivative-jump - based gost penalty stabilization wic is prevalent in te literature, see e.g. [BH12], [MLLR14], [BHL15], [BHLZ16]. Compared to te derivative-jump -based stabilization it as certain computational advantages especially for iger order metods. To regain control over cut elements, te derivative-jump -based stabilization needs to take into account te normal derivatives over te facets up to order k s as seen in equation (3.4). Tus, te computational effort increases wit te polynomial degree of te ansatz functions. Moreover, te derivativejump -based stabilization usually associates a different stabilization parameter to eac j = 0,..., k s. Ten one faces te task to determine an optimal coice of all tese parameters. Te stabilization employed in tis tesis is not affected by tese problems: It is te same for all polynomial degrees. Hence, tere are less terms to assemble tan for te derivative-jump -based stabilization if te polynomial degree exceeds one. Also tere is just a single stabilization parameter to coose. Tis makes it particularly suited for iger order metods. A furter discussion of implementational and practical aspects of derivative-jump - based and projection -based gost penalty can be found in section of [Sc17]. Lemma 3.1 allows to bound te norm of a discrete function on te extended time slab E(Q n ) by its norm on I(Q n ) and te corresponding stabilization terms. Lemma 3.2. Tere exists a constant C > 0 suc tat for every u W W tere olds ( ) u 2 2 E(Q n ) C j n γ (u, u) + u 2 I(Q n ). (3.5) J 17

23 Proof. Decompose te norm u 2 E(Q n ) into a sum over elements Qn T = T I n: u 2 E(Q n ) = t n T E(Ω n ) t n 1 t n = u(, t) 2 T dt t n 1 u(, t) 2 T dt + t n T I(Ω n ) T E(Ω n )\I(Ω n ) t n 1 u(, t) 2 T dt. If T I(Ω n ) we are done. For T E(Ω n ) \ I(Ω n ) we use assumption A.1: Tere exists a B(T ) = T 0 I(Ω n ) and elements {T i } M i=0 wit correponding facet patces contained in {ω F F F,n R } tat can be crossed in order to traverse from T M = T to T 0. We ten apply Lemma 3.1 iteratively to eac neigboring pair {T i, T i 1 } to bound te norm on T M against te norm on T 0 at te expense of stabilization terms: t n t n u(, t) 2 T dt C u(, t) 2 B(T ) dt + t n u ωf (, t) 2 ω F dt. t n 1 F F,n t n 1 t n 1 Repeating tis for every element T E(Ω n ) \ I(Ω n ) one ends up wit u 2 E(Q n ) C ( T I(Ω n ) ( 1 + #(B 1 (T )) ) tn t n 1 + #{T E(Ω n ) \ I(Ω n )} C t n u(, t) 2 T dt + T I(Ω n ) t n 1 ( ) = C u 2 I(Q n ) + C 2 B j n γ (u, u), J R u(, t) 2 T dt F F,n R F F,n R t n ) u ωf (, t) 2 ω F dt t n 1 ( 1 + t ) t n u ωf (, t) 2 ω F dt t n 1 ( ) in view of #(B 1 (T 0 )) C for T 0 I(Ω n ) and #{T E(Ω n )\I(Ω n )} C B 1 + t. Te previous Lemma allows to extend estimates for finite elements wit tensor product structure to te unfitted case at te expense of additional stabilization terms. Lemma 3.3 (Stabilized inverse inequality in time). For u W it olds tat ( u n 1 +, u n 1 + )Ω C ( ) j n n 1 t γ (u, u) + u 2 I(Q n ), J ( u n, u n C ( ) j )Ω n n t γ (u, u) + u 2 I(Q n ). J Proof. We employ te extension operator E to extend te domain Q n to a domain E(Q n ) = E(Ω n ) I n tat as tensor product structure. At a fixed point x E(Ω n ), 18

24 te function u(x, ) is a polynomial of degree k t in time. Tus, te trace inverse inequality on te time interval I n from Teorem 2 of [WH03] can be applied: u(x, t n ) 2 C(k t) u(x, t) 2 dt, t were te constant C(k t ) only depends on k t. Ten u n 2 Ω u n n 2 = u(x, t E(Ω n ) n ) 2 dx E(Ω n ) C 3.3 t C(k t ) t ( 2 E(Ω n ) I n I n u(x, t) 2 dt dx = γ J j n (u, u) + u 2 I(Q n ) ), C(k t ) t u 2 E(Q n ) were te last inequality follows by applying Lemma 3.2. Analogously for u n 1 +. Lemma 3.4 (Stabilized inverse inequality for time derivative). For u W W it olds tat t( t u, t u) Q n C ( ) j n t γ (u, u) + u 2 I(Q n ). (3.6) J Proof. For a fixed point x E(Ω n ), Teorem in [BS08] implies t t u(x, t) 2 C inv dt u(x, t) 2 dt, t I n were C inv does not depend on x, but only on k t. Integrating over E(Ω n ) and using tat E(Ω n ) I n = E(Q n ) leads to I n t( t u, t u) E(Q n ) C inv t u 2 E(Q n ). Hence, te desired estimate again follows by extending Q n product structure: to a domain wit tensor t( t u, t u) Q n t( t u, t u) E(Q n ) were te last inequality uses Lemma 3.2. C inv t u 2 E(Q n ) C ( ) j n t γ (u, u) + u 2 I(Q n ), J Note tat on W we ave u 2 C 3.3 t ( ) 2 j n γ (u, u) + u 2 I(Q n ). J 19

25 Lemma 3.5 (Derivatives in gost penalty). Tere exist constants C t 3.5 1, C s only depending on te polynomial degree and te sape regularity of te background mes T suc tat: olds for all u W. J( t u, t u) Ct 3.5 J(u, u), (3.7) t2 J( u, u) Cs 3.5 J(u, u) (3.8) 2 Proof. Te set of space-time facet patces as a tensor product structure witin eac time slab: F F,n ω F I R n. Tis allows to proceed similarly to te previous proofs. For u W one obtains by applying Teorem from [BS08] for te time integral on I n tat J( t u, t u) = Ct 3.5 t 2 ( 1 + t ) γ J = Ct 3.5 J(u, u). t2 ( 1 + t F F,n R ) γ J 1 2 ω F F F,n R I n 1 2 ω F ( t u ωf (x, t)) 2 dt dx I n ( u ωf (x, t)) 2 dt dx Te result for te gradient follows by applying Teorem from [BS08] for te integral over te facet patc ω F : J( u, u) = Cs ( 1 + t ) γ J = Cs 3.5 J(u, u). 2 ( 1 + t F F,n R ) γ J 1 2 I n F F,n R ω F 1 2 ( u ωf (x, t)) 2 dt dx I n ω F ( u ωf (x, t)) 2 dt dx Stability Te inverse estimates from te last section will be needed to derive a discrete inf-sup stability result. We begin by proving some auxiliary Lemmas. Lemma 3.6 (positiveness). For u W tere olds (a) d(u, u) = 1 N 1 ( u n 2 2 ) Ω u n n Ω n 2 u N 2 1 Ω N 2 u Ω+, 0 (b) b(u, u) = 1 2 N 1 u n 2 Ω n N 1 ( u n 2 Ω u n n 2 + Ω n ) (c) B(u, u) + J(u, u) = 1 2 u 2 + N ( u, u) Q n + J(u, u) u Ω 0,

26 Proof. (a) In te proof of Lemma 2.1 it was derived tat d(u, v) = { ( u n, v n )Ω ( ) u n 1 n +, v+ n 1 } + d (u, v). Ω n 1 Setting v = u and noting tat d(u, u) = d (u, u) yields d(u, u) = 1 2 (b) Some rewriting gives b(u, u) = 1 2 ( u n 2 ) Ω u n 1 2 n + Ω n 1 = 1 N 1 ( u n 2 ) u n 2 2 Ω n Ω n 2 N 1 ( u n, u n +) Ω n N 1 u N 2 1 Ω N 2 u 0 + ( u n, u n +)Ω n + u Ω 0 2. Ω 0 = 1 N 1 u n 2 Ω + 1 N 1 ( u n 2 n 2 + u n, u n + + u n + )Ω u 0 n + 2 = 1 N 1 u n 2 Ω + 1 N 1 ( u n 2 2 n + ) u n 2 2 Ω n + u 0 2 Ω n +. Ω 0 (c) Adding te expression obtained in (a) and (b) gives Since tis gives te result. b(u, u) + d(u, u) = 1 2 N 1 = 1 2 u 2. u n 2 Ω n B(u, u) = b(u, u) + d(u, u) + u Ω u N Ω N ( u, u) Q n Ω 0 Lemma 3.7 (control on time derivative). For u W tere olds wit were B(u, t t u) + J(u, t t u) 1 2 C 3.7 t( t u, t u) Q n C 3.7 u 2 N ( u, u) Q n C 3.7 J(u, u), C 3.7 = max{ C 2 3.3, 4γ J t + C C t 2γ 3.5, C C o w 2 } J C 3.4 = C 3.4 C s 3.5 and C 3.3 = C 3.3 C t

27 Remark 7. Under assumption A.3, i.e. 2 C G t, te constant C 3.7 can be bounded independently of t and. Proof. Consider B(u, t t u) t( t u, t u) Q n = (w u, t t u) Q n I N 1 ( + u n, t( t u) +) n + ( ) u 0 Ω +, t( n t u) ( u, ( t t u)) Q n. Eac of te terms on te rigt and side will be treated separately. First we always apply te Caucy-Scwarz inequality. Ten Young s inequality for some ε j > 0 will be used. We start wit II = N 1 ( u n, t( t u) n +) ε 1 2 u ε 1 ε 1 2 u 2 + C 3.3 2ε 1 ε 1 2 u 2 + C 3.3 2ε 1 ε 1 C 2 u ε 1 Ω n + ( u 0 +, t( t u) 0 + ) Ω 0 t 2( ) ( t u) n 1 +, ( t u) n 1 + Ω n 1 ( ) t 2 J( t u, t u) + t ( t u, t u) γ Q n J ( ) 2 γ J t Ct 3.5J(u, u) + t ( t u, t u) Q n ( 2 N ) γ J t J(u, u) + t ( t u, t u) Q n, were Lemma 3.3 was used in line tree and Lemma 3.5 in line four. Here C 3.3 = C 3.3 C3.5. t For te diffusion term one can use Lemma 3.4 and Lemma 3.5 to estimate wit III = ( u, ( t t u)) Q n ε 2 2 ( u, u) Q + 1 2ε 2 t 2 C 3.4 = C 3.4 C3.5. s Coosing ε 2 = C 3.4 leads to: ( t u, t u) Q n ε 2 2 ( u, u) Q + C ( ) J( u, u) + ( u, u) 2ε 2 γ Q J ε 2 2 ( u, u) Q + C ( ) J(u, u) + ( u, u) 2ε 2 γ Q J ( u, ( t t u)) Q n C 3.4 ( u, u) Q + 2γ J C 3.4 J(u, u). Ω 0 II III 22

28 It remains to treat te convection term: I = (w u, t t u) Q n ε N 3 2 t (w u, w u) Q n + t ( t u, t u) 2ε Q n 3 ε 3 2 t w 2 ( u, u) Q + t ( t u, t u) 2ε Q n. 3 Finally, te gost penalty term is estimated by invoking Lemma 3.5: were ε 4 = C t 3.5 was cosen. Assembling all te estimates gives: J(u, t t u) ε 4 t2 J(u, u) + J( t u, t u) 2 2ε 4 ε 4 2 J(u, u) + Ct 3.5 2ε 4 J(u, u) = C t 3.5J(u, u), B(u, t t u) + J(u, t t u) ( t t u, t u) Q [ J(u, u) Coosing ε 1 = 2 C 3.3 and ε 3 = 2 yields: ( u, u) Q n B(u, t t u) + J(u, t t u) 1 2 ( t tu, t u) Q 1 1 2ε 3 [ C ε 1 γ J t + [ C 3.3 2ε 1 ] C C3.5 t 2γ J C ε 3 2 t w 2 C 3.3 u 2 J(u, u) [ ] ( u, u) Q C t w 2. Using assumption A.4, te claim follows wit C 3.7 = max{ C 3.3, 2 4γ J t + C 3.4 2γ J + C t 3.5, [ ] ] 2 4γ J t + C C o w 2 }. ε 1 2 u 2. C ] C3.5 t 2γ J Lemma 3.8 (special function). For u in W tere olds t t u j C 3.8 u j, wit (C 3.8 ) 2 = max{c C 3.3, 2 γ J t Ct 3.5[C C 3.3 ] + C 3.4 γ J C s C t 3.5}. Remark 8. Te constant C 3.8 can be bounded independently of t and if assumption A.3 is used. 23

29 Proof. Most of te estimates ave already been used in te previous proof. From Lemma 3.4 and Lemma 3.5 it follows tat t 3( t 2 u, t 2 u ) ( ) 2 C Q 3.4 t J( t u, t u) + ( t u, t u) γ Q J Wit Lemma 3.3 and Lemma 3.5 we ave: 2 C 3.4 C3.5 t γ J t J(u, u) + C 3.4 t( t u, t u) Q. N [( ) t 2 t u 2 2 t 2 ( t u) + n 1, ( t u) + n 1 + ( ] ( Ω n 1 t u) n, ( t u) n )Ω n ( ) 2 4C 3.3 t J( t u, t u) + ( t u, t u) γ Q J 2 4C 3.3 C3.5 t γ J t J(u, u) + 4C 3.3 t( t u, t u) Q. Anoter application of Lemma 3.4 and Lemma 3.5 yields ( ) t 2 2 ( t u, t u) Q C 3.4 J( u, u) + ( u, u) γ Q J C3.5 s C 3.4 J(u, u) + C 3.4 ( u, u) γ Q. J Finally, te Gost-Penalty term is bounded by Lemma 3.5: C3.5J(u, t u). Collecting all te terms gives: t t u 2 j (C C 3.3 ) t( t u, t u) Q n + C 3.4 N t 2 J( t u, t u) ( u, u) Q n ( 2 + γ J t Ct 3.5[C C 3.3 ] + C ) 3.4 C3.5 s + C3.5 t J(u, u) γ J (C 3.8 ) 2 u 2 j, wit (C 3.8 ) 2 = max{c C 3.3, 2 γ J t Ct 3.5[C C 3.3 ] + C 3.4 γ J C s C t 3.5}. Te previous Lemmas can now be combined to sow discrete stability. Proposition 3.9 (Stability). For all u W tere exists a ṽ(u) W suc tat tere olds B(u, ṽ(u)) + J(u, ṽ(u)) C 3.9 u j ṽ(u) j, wit C 3.9 = 1 2(2C 3.7 +C ). Remark 9. Consider te constants C 3.7 (γ J ), C 3.8 (γ J ) as a function of te gost-penalty stabilization parameter. Ten lim C 3.7(γ J ) =, γ J +0 lim C 3.8(γ J ) =, wic implies γ J +0 lim C 3.9(γ J ) = 0. γ J +0 Tat is, te stability estimate deteriorates as γ J decreases since we loose control over te norms on te elements of te extended domain. 24

30 Remark 10. If one cooses not to include te stabilization factor ( ) 1 + t in te gost penalty term ten it appears as 1/ ( ) 1 + t in front of te constant C3.9 in te stability estimate. Proof. Consider ṽ(u) = (2C )u + v (u) W wit v (u) = t t u W. Using te triangle inequality and Lemma 3.8 tere olds Hence wit Lemma 3.6 and Lemma 3.7: B(u, ṽ(u)) + J(u, ṽ(u)) ṽ(u) j (2C ) u j + v (u) j (2C C ) u j. = (2C ) [B(u, u) + J(u, u)] + B(u, t t u) + J(u, t t u) ( ) 1 (2C ) 2 u 2 + ( u, u) Q n + J(u, u) + 1 t( t u, t u) 2 Q n ) C 3.7 ( u 2 + ( u, u) Q n + J(u, u) ( = 1 N ) 2 u 2 + (1 + C 3.7 ) ( u, u) Q n + J(u, u) u 2 j 1 2(2C C ) u j ṽ(u) j. t( t u, t u) Q n Continuity Te continuity of te bilinear form B(, ) wit respect to te first argument is needed in a non-discrete space, since it as to be evaluated later at te solution of te continuous problem. Proposition 3.10 (Continuity). For all u W + H 1 (Q) and v W tere olds for C 3.10 = 4 + w Co. B(u, v) C 3.10 u v, J(u, v) u J v J, Proof. Here one can use te representation B(u, v) = d (u, v) + b (u, v) + a(u, v) obtained in Lemma 2.1. We bound te terms one after anoter: 25

31 By te Caucy-Scwarz inequality and te boundedness assumption on w : d (u, v) = (u, t v w v) Q n ( ) 1 ( ) 1 t u, u ( t t v, t v) Q n + Q t u, u ( tw v, w v) Q n n Q n N ( ) 1 t u, u N ( t t v, t v) Q n + w t N ( v, v) Q n Q n ) (1 + w Co u v. For u H 1 (Q) te inverse inequality from Lemma 3.3 is not available. Tus, te term u as been included into te -norm. b (u, v) N 1 (u n, u n ) Ω n ( v n, v n ) Ω n + (u N, u N ) Ω N (v N, v N ) Ω N N 1 (u n, u n N 1 ) Ω n ( v n, v n ) Ω n + u N 1 ( v n, v n ) Ω n + (v N, v N ) Ω N 2 u v. (u N, u N ) Ω N (v N, v N ) Ω N Te diffusion term is easily bounded: a(u, v) N ( u, u) N Q n ( v, v) Q n u v. Te stabilization term can be bounded by repeated application of te Caucy- Scwarz inequality: ( J(u, v) = 1 + t ) 1 γ J u 1 ω F v ω F ( 1 + t ( 1 + t ) γ J ) γ J ( 1 + t ) γ J F F,n R F F,n R ω F I n ω F I n F F,n R ω F I n F F,n R ω F I n 1 2 u 2 ω F ω F I n 1 ( 2 u 2 ω F 1 2 u 2 ω F 1 + t 1 2 v 2 ω F ) γ J ( 1 + t ) γ J F F,n R ω F I n F F,n R ω F I n 1 2 v 2 ω F 1 2 v 2 ω F = J(u, u) J(v, v) = u J v J. 26

32 3.2.4 Consistency Proposition 3.11 (consistency error). Let u H 2 (Q) be te solution of (2.1) and u be te solution of (2.7). Ten tere olds: B(u u, v ) J(u, v ) = 0 for all v W. Proof. Let v W. From u H 2 (Q) it folllows tat u L 2 (Q) and u n = 0 for n = 1,..., N 1 in te L 2 (Ω n ) sense. Tus, b(u, v) = ( ) u 0, v+ 0. Ω 0 Moreover, integration by parts of te diffusion term leads to a(u, v) = N ( u, v) Q n due to omogeneous Neumann boundary conditions. Hence, B(u, v) = d(u, v) + b(u, v) + a(u, v) = ( t u + w u u, v) Q n + ( ) u 0, v+ 0 = (f, v) Q n + ( ) u 0, v+ 0 = f(v). Ω 0 Subtracting from tis te equation for te discrete solution yields te claim Céa-like result B(u, v) + J(u, v) = f(v) Teorem 3.12 (Céa-like result). Let u H 2 (Q) be te solution of (2.1) and u be te solution of (2.7). Ten tere olds: u u wit C 3.12 = C 3.10 (C 3.9 ) 1. inf w W ( u w + C 3.12 u w + (C 3.9 ) 1 w J ), (3.9) Remark 11. Te first two terms in te estimate (3.9) above describe te approximation error of te solution u in te space W wit respect to te norms and. Denote by w te best approximation to u in W. Ten te last term in (3.9) measures te semi-norm w J of te best approximation rater tan te approximation error u w itself. Tis is reasonable, since we ave defined J(, ) only for discrete functions. An estimate for w J will be sown in Proposition Te key observation for obtaining tis estimate is te possibility to bound te jump on te facet patces ω F from above by te approximation error of an L 2 -projection P ωf on te facet patc: w (, t) ωf C (w P ωf w )(, t) ωf. Proof. Coose an arbitrary w W and split te error into two parts: u u e u + e w, 27 Ω 0

33 wit e u = u w and e w = w u. Combining Propositions 3.9, 3.10 and 3.11 from above ( and introducing q = ṽ(e w ) wit ṽ( ) as in Proposition 3.9) yields e w q j e w j q j (C 3.9 ) 1 (B(e w, q) + J(e w, q)) It follows tat: = (C 3.9 ) 1 (B(w, q) + J(w, q) B(u, q) J(u, q)) = (C 3.9 ) 1 (B(w, q) + J(w, q) B(u, q)) = (C 3.9 ) 1 (B( e u, q) + J(w, q)) C 3.10 (C 3.9 ) 1 e u q j + (C 3.9 ) 1 w J q j. e w C 3.10 (C 3.9 ) 1 e u + (C 3.9 ) 1 w J. As w may be cosen arbitrarily te result follows. 3.3 Interpolation in space-time In order to derive an a priori estimate from Teorem 3.12 an interpolation operator I Γ mapping H k (Q) into te space-time finite element space W is needed. Te global interpolation operator will be defined by its restriction to te time slabs IΓ n : Hk (Q n ) W n. So te task is to construct IΓ n. Tis operator will be build step by step. In subsection we first construct an interpolation operator Π n W : L2 (E(Q n )) W n. Here te space-time domain E(Q n ) = E(Ω n ) I n as a tensor product structure. In order to keep te notation sort we denote Q n = Ω I n in tis subsection wit a generic polyedral domain Ω. We may tink of Ω as te extended spatial domain wic combined wit I n forms te extended space-time slab. Te operator Π n W is obtained by concatenation of purely temporal and purely spatial interpolation operators. Consequently, te first task will be te construction of tese operators. If te operator Π n W for tensor product domains wit te desired approximation properties is available, ten IΓ n can be defined by means of assumption A Interpolation in tensor-product space-time spaces In tis section we consider space-time slabs wit tensor product structure Q n = Ω I n. In [Le15] suitable space-time interpolation operators for te piecewise linear case ave been constructed. Te aim of tis section is to generalize tese results. Anisotropic Sobolev spaces We introduce anisotropic Sobolev spaces (see eg. [BIN78],[WYW06]) on te space-time domain Q in wic temporal and spatial derivatives are treated differently. Due to te special role of te time, tese spaces are also called t-anisotropic Sobolev spaces. Define: H k,l (Q) := {u p t D α x u L 2 (Q), p, q N, q = α, q k + p l 1}. (3.10) For k = l te isotropic Sobolev spaces are retained: H k,k (Q) = H k (Q). Moreover, tere exists a (time) trace operator H 0,1 (Q n ) L 2 (Ω), wic is bounded: u(, t) Ω C u H 0,1 (Q n ). 28