ERROR ESTIMATES FOR THE DISCONTINUOUS GALERKIN METHODS FOR PARABOLIC EQUATIONS. 1. Introduction. We consider the parabolic PDE of the form,

Transcription

1 ERROR ESTIMATES FOR THE DISCONTINUOUS GALERKIN METHODS FOR PARABOLIC EQUATIONS K. CHRYSAFINOS AND NOEL. J. WALKINGTON Abstract. We analyze te classical discontinuous Galerkin metod for a general parabolic equation. Symmetric error estimates for scemes of arbitrary order are presented. Te ideas we develop allow us to relax many assumptions freqently required in previous work. For example, we allow different discrete spaces to be used at eac time step and do not require te spatial operator to be self adjoint or independent of time. Our error estimates are posed in terms of projections of te exact solution onto te discrete spaces and are valid under te minimal regularity guaranteed by te natural energy estimate. Tese projections are local and enjoy optimal approximation properties wen te solution is sufficiently regular. 1. Introduction. We consider te parabolic PDE of te form, u t + Atu = F t, u = u. 1.1 Te operators act on Hilbert spaces related troug te standard pivot construction, U H H U, were eac embedding is continuous and dense. Ten, A. : U U is a linear map and F. U. Our goal is to analyze te classical discontinuous Galerkin DG sceme and derive fully-discrete error estimates under mimimal regularity assumptions. Te class of DG scemes we consider are classical in te sense tat te discrete solutions may be discontinuous in time but are conforming in space, i.e. are in a subspace of U at eac time. Our tecniques also apply to te more general implicit evolution equation [22, 23] Mtu t + Atu = F t, u = u, 1.2 were M. : H H is a self adjoint possitive definite operator. Te extension of our analysis to tis equation will be taken up seperatly. Te analysis below addresses te following issues wic ave not yet been adequately considered in te literature. Te operator A. may depend upon time and is not required to be self adjoint. To date te sarpest estimates for DG approximations exploit classical spectral teory for self adjoint positive definite operators, so require A to be suc an operator and to be independent of time. Wen A. is not self adjoint, multiplying 1.1 by u t does not give an estimate for te time derivative. Te subspaces of U used for te DG approximations may be different on eac time interval, t n ]. Tis adds a significant complication to te analysis Matematics subject classification: 65M6 Department of Matematics, Carnegie Mellon University, Pittsburg, PA Supported in part by National Science Foundation Grants DMS 28586, and ITR Tis work was also supported by te NSF troug te Center for Nonlinear Analysis. 1

2 wic is present even wen A =. Indeed, te first step in our analysis is to consider te DG sceme for ordinary differential equations in te Hilbert space H. Different subspaces are an essential ingredient of adaptive strategies used in conjunction wit a-posteriori error estimates to give guaranteed error bounds. Retriangulation is also necessary for many algoritms based upon a Lagrangian coordinate system; below we present suc an example. DG approximations of equations of te form 1.2 ave not been considered in te past. Below we sow tat equations of tis form arise wen Lagrangian scemes are constructed for te convection diffusion equation [6, 7, 8]. Te operator A. is not required to be strictly coercive; we only require an assumption of te form A.u, u c u 2 U C u 2 H. Here. U is a semi-norm suc tat. 2 U =. 2 U +. 2 H. Tis causes significant problems in te analysis of DG scems since te classical Gronwall argument, used for te continuous problem, fails in te discrete setting. Tis failure is due to te elemenary observation tat functions of te form χ u are not polynomial in time unless [,ˆt ˆt is a partition point, so are not available as test functions in te discrete setting. In te past tis problem as been circumvented by bounding temporal derivatives of te solution [5, 24] so tat te solution between te partition points can be controlled by te values at tese points. Tis line of argument fails for solutions aving minimal regularity. Below we circumvent tese issues by constructing polynomial approximations to te caracteristic functions χ. [,ˆt As stated above, our analysis does not require any regularity above and beyound te natural bounds tat follow from te usual energy estimate. Tis is essential for control problems were solutions of te dual problem typically will not enjoy any additional regularity. Our estimates sow tat te error can be bounded by te local truncation error of te ordinary differental equation obtained by setting A =. Tese local truncation errors can also be viewed as te approximation error of local projections of te solution onto te discrete subspaces. In our analysis we are careful to keep track of ow te various constants depend upon te coercivity constant on A.. Tis is important for te analysis of problems like te convection diffusion equation were te coercivity constant is small. We present and example of equations wic can be analyzed witin te general framework developed ere but fall outside of te teory developed, for example, in Tomée s text [24]. Convection Diffusion Equation: Te classical convection diffusion equation is ū t + V. ū ɛ ū =, and te problems tat arise wen ɛ is small are notorious. To address tese problem tis equation is sometimes considered in a Lagrangian variable. Specifically, let Ṽ be a Here χ [,ˆt is te caracteristic function equal to 1 on [, ˆt and zero oterwise. 2

3 numerical approximation of V and let x = χt, X be te cange of variables defined by te flow map associated wit Ṽ, i.e. If ut, X = ūt, xt, X ten ẋt, X = Ṽt, xt, X, x, X = X. u t + V Ṽ.F T X u ɛ1/jdiv X JF 1 F T X u =, were F ij = x i / X j is te Jacobian of te mapping and J = detf. Te natural weak problem for tis equation is ut + V Ṽ.F T X u v + ɛf T X u.f T X v J =. Ω Using te properties of determinants we find u t J = Ju t J u = Ju t J divṽu. If divṽ = ten J is constant te transformed problem takes te form of 1.1; oterwise, it takes te form of 1.2 wit M.u = Ju, and A.u = divṽuj + V Ṽ.F T X uj ɛ div X JF 1 F T X u. Tis statement of te problem generalizes te idea beind te caracteristic Galerkin sceme introduced by Douglas and Russel in [6] and DuPont in [7]. Tis cange of variables reduces te effective Peclet number, V Ṽ /ɛ, to order O1 if Ṽ is a sufficiently accurate approximation of V. Tis will eliminate many of te numerical difficulties encountered by algoritms based upon te classical statement; owever, oter problems arise. Wile te Jacobian of te transformation satisfies F, X = I, its condition number grows exponentially if Ṽ is anyting oter tan a rigid motion. In te context of a numerical sceme tis problem is circumvented by reinitalizing te transformation at eac or every few time steps. Tis reinitialization corresponds to canging te subspace for te numerical solution every few time steps. In essence, a trianglular mes in te X coordinate system will be a distorted mes in te x-coordinate system, and reinitializing te transform corresponds to projecting te solution onto a straigt sided triangular mes in te x coordinates. Tis gives rise to different subspaces at eac time step Related Results. Te discontinuous Galerkin metod was first introduced to model and simulate neutron transport by Lasaint and Raviart in [15]. Tere is an abundant literature concerning applications of te DG sceme in yperbolic problems, see e.g. [4, 14, 25] and references witin. Te DG metod for ordinary differential equations was considered by Delfour, Hager and Trocu in [5]. Tey sowed tat te DG sceme was super convergent at te partition points order 2k+2 for polynomials of degree k. Te super convergence results and better rates in te H norm use a duality argument, and space considerations do not permit us to develop te corresponding estimates in te current setting. 3

4 In te context of parabolic equations DG scemes were first analyzed for linear parabolic problems by Jamet in [13] were Ok q results were proved and by Eriksson, Jonson and Tomée in [11] were Ok 2q 1 results were proved for smoot initial data among oters. An excellent exposition of teir results and, more generally, te DG metod for parabolic equations, can be found in Tomée s book [24]. In [24] nodal and interior estimates are presented in various norms. One may also consult [18] for te analysis of a related formulation based on te backward Euler sceme. Te relation between te DG sceme and adaptive tecniques was studied in [9] and [1]. Finally, some results concerning te analysis of parabolic integro-differential equations by discontinuous Galerkin metod are presented in [16] see also references terein. In [8] DuPont and Liu introduce te concept of symmetric error estimates for parabolic problems. Tey define suc an error estimate to be one of te form u u C inf u w, w U were u and u are te exact and approximate solutions respectively,. is an appropriate norm, and U is te discrete subspace in wic approximation solutions are sougt. Wile estimates of tis form are standard for elliptic problems, tis is not te case for evolution problems. For example, error estimates for evolution problems approximated by te implicit Euler sceme frequently involve terms of te form u tt L 2 Ω. Symmetric error estimates are useful for problems were te solution u may not be very regular, suc as control problems, and are used to develop a-posteriori error estimates for adaptive scemes. Symmetric error estimates for moving mes finite element metods were studied in [8, 17] see also references witin. Mes modification tecniques for finite elements ave also been introduced in [19] and [2]. For some earlier work on convection-dominated problems based on te metods of caracteristics and mes modification one may consult [6] and [7] respectively. An alternative to te symmetric error estimates are estimates of te form u u C u P u, 1.3 were P : U U is a projection wic exibits optimal interpolation properties if u is sufficiently smoot. Estimates of tis form enjoy te same advantages of tose proposed by DuPont and Liu. Below we construct an estimate of te form 1.3 for parabolic equations of te form 1.1; were te projection P u is te numerical approximation of an ODE, so is not local. However, u P u u P loc u + P u P loc u, were P loc is a local projection, so te first term can be estimated using classical interpolation teory. Te second term P u P loc u vanises if te same subspace of U is used in eac partition, t n ; oterwise, it depends solely upon te jump in te interpolant of te exact solution at te partition points {t n } N n=. Te size of te constant C in 1.3, and its dependence on various constants, play an important role; below we are careful to state te dependence of te constant upon te various coercivity constants and bounds assumed for te operator A. 4

5 Error estimates for Lagrange-Galerkin approximations of convection dominated problems for divergence-free velocity fields vanising on te boundary are presented in [3]. Issues related to te stability of Lagrange-Galerkin approximations are also discussed in [21]. Recently tere as been a lot of work on te development and analysis of discontinuous Galerkin metods for elliptic problems. A compreensive survey and comparison of tis work can be found in [2] wic contains many references related to tis approac Outline. In Section 2, we formulate and analyze te DG sceme for te ordinary differential equation corresponding to A.. In tis section we focus on te difficulties tat arise wen different subspaces of U may be used at every time step. Error estimates are first derived at times corresponding to te partition point. Additional arguments using discrete caracteristic functions are developed to estimate te error and at times inbetween te partition points. Te arguments we use appear to be new. A priori estimates for te DG approximations of 1.1 are developed in Section 3. Estimates are derived in te natural norms associated wit te parabolic problem; by natural we mean norms tat arise in te natural energy estimates obtained by multiplying 1.1 by u. Te results of Section 2 are used in an essential fasion. Indeed, te difficulties associated wit different subspaces of U at eac time step are circumvented by comparing te discrete solution of te parabolic equation wit and appropriate solution of an ODE. By using te discrete caracteristic functions developed in Section 2 we can avoid te self adjoint assumptions typically imposed upon A.. Wen te same discrete subspace of U is used for eac time step our tecniques generalize, and to some extend simplify, te classical analysis. Te reader interested in tis case only needs to read Sections 2.3 and 2.5 on te construction of discrete caracteristic functions, and Definition 2.2 for P loc from Section 2 before proceeding to Section 3. Remark 4 in Section 3 amplifies upon tis Notaton. Trougtout we assume tat te evolution of te solution to 1.1 takes place in a Hilbert space H and te operators A. are defined on anoter Hilbert space U wit U H H U, were eac of te embeddings are dense and continuous. Te inner product on H is denoted by.,. and te induced duality pairing between U and U will be dentoted by.,.. Te norm on H is often denoted by.. H, and we assume tat te norm on U can be written as. 2 U =. 2 U +. 2 H were. U is a semi-norm on U te principle part and is often denoted by. ;. 2 U = Standard notation of te form L 2 [, T ; U], H 1 [, T ; U ] etc. is used to indicate te temporal regularity of functions wit values in U, U etc. Approximations of 1.1 will be constructed on a partition = t < t 1 <... < t N = T of [, T ]. On eac interval of te form, t n ] a subspace U n of U is specified, and te approximate solutions will lie in te space U = {u L 2 [, T ; U] u,t n ] P k, t n ; U n }. Here P k, t n ; U n is te space of polynomials of degree k or less aving values in U n. Notice tat, by convention, we ave cosen functions in U to be left continuous 5

6 wit rigt limits. We will write u n for u t n = u t n, and let u n + dentote ut n +. Tis notation will is also used wit functions like te error e = u u. We always assume te exact solution, u, is in C[, T ; H] so tat te jump in te error at t n, dentoted by [e n ] is equal to [u n ] = u n + u n. 2. DG sceme for an ODE Background. In tis section we address te issues tat arise wen different discrete subspaces are used for eac step of a time of te DG sceme. It suffices to consider suc DG scemes in te context of an ODE in a Hilbert space since te additional terms appearing in te error estimates are te same for bot te ODE and parabolic PDE case. Also, te solution of te ODE will be used to obtain error estimates for te parabolic PDE under minimal regularity assumptions. We consider te problem of recovering a function u C[, T ; H] H 1 [, T ; U ] given te initial value u and its derivative f = u t. Specifically, we consider te DG finite element approximations for te initial value problem u t = f, u = u. 2.1 Recall tat H and U are related troug a pivot space construction, U H H U. In tis situation tere exists a unique u C[, T ; H] H 1 [, T ; U ] wic is te solution of te weak problem, ut, vt T u, v t = u, v + v C[, T ; H] H 1 [, T ; U ]. T f, v 2.2 Recall tat we write for te norm H and write te inner product in H as.,.. We also write u 2 U = u 2 + u 2 were. is a semi-norm on U te principle part. Te U, U duality pairing is denoted by, To approximate te solution of 2.2 we introduce a partition = t < t 1 <... < t N = T of [, T ] and a collection {U n}n n= of subspaces of U. Te DG metod constructs an approximate solution suc tat u U {u L 2 [, T ; H] u,t n ] P k, t n ; U n } tn u n, v n u, v t u n 1, v+ n 1 = f, v, 2.3 for all v U and eac n = 1, 2,... N. Recall tat u n u t n = u t n and we employ te standard notation u n +, u n for te traces from above and below respectively. Integration by parts gives te following alternative form of 2.3 u t, v + u n 1 + un 1, v n 1 + = tn f, v

7 2.2. Error Estimate at Partition Points. In tis elementary context it is possible to explicitly write an expression for te error at eac nodal partition point t n, see equation 2.5 below. A simple consequence of tis formula is te following teorem wic provides a decomposition of te error into te errors due to te canging of te spaces and te initial projection error. Teorem 2.1. Let u U be te approximate solution of 2.1 computed using te discontinuous Galerkin sceme 2.3and let P n : H U n denote te projection operator in H, and write ê n = P n ut n u n. Ten ê n = n n P j I Pi 1 ut i 1 + P n P n 1... P ê j=i In particular, ê n n P i I P i 1 ut i 1 + ê. 2.6 Remark 1. 1 Note tat P i I P i 1 = wen U i U i 1, so ê n ê wen te same discrete subspace is used at eac time. 2 If e n = ut n u n is te total error at time t n ten e n I P n ut n + ê n. Wen te same space is used at eac step te first term becomes I P ut n wic is useful only if ut n can be well approximated in U at all times. Wen tis is not te case, an ideal strategy would cose te spaces {U n } to track te solution so tat bot I P n ut n and te jump terms in equation 2.6 are small. Proof. Let e = u u be te total error and note tat te Galerkin ortogonality gives, e n, v n e, v t e n 1, v+ n 1 =. 2.7 If v t v n is independent of time, ten te middle term vanises to give e n, v n = e n 1, v n, v n U n. It follows tat so tat ê n = P n e n ê n = P n e n 1 = P n u u n 1 = P n u ± P n 1 u u n 1 = P n I P n 1 u + P n ê n 1. Using te above relation inductively yields

8 2.3. Discrete Caracteristic Functions. To compute te error at arbitrary times t [, t n we would like to substitute v = χ [,tu into equation 2.4 were χ [,t is te caracteristic function on [, t. Clearly tis function is not in U so in tis section we construct discrete approximations of suc caracteristic functions to circumvent tis problem. Te construction of te discrete caracteristic functions is invariant under translation so it is convinient to work on te interval [, τ wit τ = t n. We begin by considering polynomials p P k, τ. A discrete approximation of χ [,t p is te polynomial p { p P q, τ p = p} satisfying pq = t pq q P k 1, τ. Te above construction is motivated by te fact tat we may put q = p to obtain τ p p = t pp = p 2 t p 2. We next extend tis elementary construction to approximate functions of te form χ [,t v for v P k, τ; U were U is any subspace of H. If v P k, τ; U we can write v = k i= p itv i were {p i } P k, τ and {v i } U. If we define ṽ = k i= p itv i it is clear tat ṽ P k, τ; U satisfies ṽ = v, and ṽ, w = t v, w w P k 1, τ; U. 2.9 In te ODE setting we could ave directly defined ṽ directly from equation 2.9 instead of te two stage construction given ere. However, for parabolic equations it is useful to observe tat v is independent of te coice of te space U Error Estimates at Arbitrary Times. To estimate te error at an arbitrary time t [, t n we use te projection operator P loc n introduced in [11]. Definition Te projection P loc n P loc n u n = P n ut n, and : C[, t n ; H] P k, t n ; U n satisfies u P loc n u, v =, v P k 1, t n ; U n. Here we ave used te convention P loc n u n P loc n ut n and P n : H U n projection operator onto U n H. 2 Te projection P loc P loc : C[, T ; H] U satisfies u U and P loc u,t n ] = P loc n u [,t n ]. is te Tis projection satisfies te te standard approximation properties, [24, Teorem 12.1], and can be tougt of as te one step DG approximation of u t = f on te interval, t n ] wit exact inital data u. To see tis, write u = P loc n u and write test 8

9 te function as te derivative v t of a function v P k, t n ; U n. Integration by parts ten gives u n, v n u, v t u, v n 1 + = tn u t, v wic is identical in form to 2.3. For te parabolic problem we will use te analogus global projection, P below wic is te DG approximation of u t = f on all of [, T ]. We are now ready to prove te main result of tis section wic sows tat te error estimate of Teorem 2.1, wic eld at te discrete times, olds for every time. Teorem 2.3. Let u U be te approximate solution of 2.1 computed using te discontinuous Galerkin sceme 2.3. Let ê = P loc u u were P loc is te projection defined in Definition 2.2. Ten and êt n P i I P i 1 ut i 1 + ê, t, t n ], [ê n 1 ] n P i I P i 1 ut i 1 + ê. were [ê n 1 ] = ê n 1 + ên 1 is te jump in ê at. More generally, if.,. V êt V ê n V = for t, t n ]. is a semi inner product on U n ten n n P j I Pi 1 ut i 1 + P n P n 1... P ê 1 V, j=i Remark 2. 1 In Teorem 2.1 we used te notation ê n to denote P n e n = P n et n. Tis is consistant wit te notation for êt above, since by construction P loc n e n = P n e n. 2 Notice tat discreteness plays an essential role in te last inequality. We ave not assumed, for example, tat u C[, T ; U], yet te last inequality gives an expression for estimate te error of P n u in tis norm. Proof. Recalling te Galerkin ortogonality relation 2.7 and applying te definition of te projection P loc n sows tat or equivalently, ê n, v n ê, v t e n 1, v n 1 + =, ê t, v + ê n 1 + en 1, v n 1 + =. Define z P, t n ; U n to be te discrete Laplacian of ê; tat is, for eac t [, t n ] z t, w = êt, w V, for all w U n. 9

10 Ten set v = z χ [,tz to be te approximate caracteristic function of z constructed in Section 2.3. Tis coice of test function gives 1/2 êt 2 V ê n V + ê n 1 + P en 1, ê n 1 + V =, so tat 1/2 êt 2 V + 1/2 ê n V P n e n 1, ê n 1 + V =. 2.1 From equation 2.8 we find ê n = P n e n 1, so an application of te Caucy Scwarz inequality and ab 1/2a 2 + b 2 sows êt V ê n V as claimed. To establis te bound on te jump term we set V = H in 2.1 and rearrange te terms to get 1/2 êt 2 + 1/2 ê n ê n 1, ê n 1 + = I P n 1u, ê n 1 +. Next let t + and complete te square on te left to obtain 1/2 ê n /2 ê n 1 + ên 1 2 = 1/2 ê n I P n 1 u, ê n 1 +. It follows tat ê n 1 + ên 1 2 ê n P n I P n 1 u 2 ê n 1 + P n I P n 1 u 2, and again we use te estimate establised in Teorem 2.1 to complete te proof. Te following definition and corollary provide a concise synopsis of te results of tis section in a form useful for te analysis of te parabolic problem in te next section. Definition 2.4. Te projection P : C[, T ; H] H 1 [, T ; U ] U is te discontinuous Galerkin approximation of te function reconstructed from it s derivative and initial data. Tat is, if u = P u, ten u is te solution of 2.3 were f = u. Te previous teorem can ten be interprated as an estimate of te difference between te global projection P u and te local projection P loc u. Corollary 2.5. Let u C[, T ; H] H 1 [, T ; U ], ten and n P u P loc ut P i I P i 1 ut i 1 + ê, t, t n ], [P u P loc un 1 ] n P i I P i 1 ut i 1 + ê, were ê = P u u. More generally, if.,. V P u P loc ut V is a semi inner product on U n ten n n P j I Pi 1 ut i 1 + P n P n 1... P ê 1 V, j=i 1

11 for t, t n ]. Remark 3. An alternative to using te last expression to estimate te error in oter norms is to postulate an inverse inequality. For example, if v C inv v for all v n U n, ten for t tn 1, t n ] P u P loc ut C inv n P i I P i 1 ut i 1 + ê. If te solution as sufficent regularity projection errors in te different norms typically satisfy C inv e e Estimates for Discrete Caracteristic Functions. Our construction of te discrete caracteristic functions in Section 2.3 was purely algebraic and, for te ODE, no bounds were required for te mapping e ẽ. Te analysis of te parabolic problem requires te bounds developed next. In te next two lemmas, te function p p will refer to te map constructed in Secton 2.3. Lemma 2.6. Te mapping p p in P k, τ is linear, continuous and tere exists a constant Ĉk depending only upon k suc tat p p L 2,τ Ĉk p 2 L 2 t,τ. Moreover, p χ [,t p L 2,τ p p L 2,τ + p χ [,t p L 2,τ 1 + Ĉk p L 2 t,τ and p L 2,τ 1 + Ĉk p L 2,τ. Proof. Since p = p we can write p p = t p wit p P k 1, τ. Ten using te definition of p, t pq = p pq = pq q P k 1, τ. t Setting q = p we obtain, c k τ p 2 t p 2 = p p, t were we use te equivalence of norms on P k, and scale to make c k independent of τ. It follows tat te matrix corresponding to te bilinear form is non-singular, so solutions exist. Te Caucy-Scwarz inequality gives wic implies, c 2 k p p 2 = c 2 k c k τ 2 p 2 t p 2, t 2 p 2 c 2 τ 2 p 2 t p 2. 11

12 We next consider te induced mapping v ṽ on P k, τ; V were V is any semi inner product space. Recall tat if v = k i= p itv i ten ṽ = k i= p itv i. Since tis construction is purely algebraic it follows tat ṽ P k, τ; V satisfies ṽ = v and ṽ, w V = t v, w V w P k 1, τ; V Lemma 2.7. Let V be a semi-inner product space, ten te mapping k i= p itv i k i= p itv i on P k, τ; V is continuous in L 2 [,τ;v ]. In particular, ṽ L 2 [,τ;v ] C k v L 2 [,τ;v ], and ṽ χ [,t v L 2 [,τ;v ] C k v L 2 [,τ;v ], were C k = k + 1 1/2 1 + Ĉk. Proof. Witout loss of generality write v = k i= p itv i were {p i } form an ortonormal basis of P k, τ in L 2, τ, so tat v 2 L 2 [,τ;v ] = k i= v i 2 V. Te lemma ten follows by direct computation: ṽ 2 V = k i,j= p i t p j tv i, v j V k p i L 2,τ p j L 2,τ v i V v j V i,j= 1 + Ĉk 2 Te second estimate follows similarly. k v i V v j V i,j= k 1 + Ĉk 2 k + 1 v i 2 V 1 + Ĉk 2 k DG Sceme for Parabolic PDE s. i= v 2 V Formulation of te DG sceme.. We turn our attention on te approximation of 1.1 using te discontinuous Galerkin sceme. In order to derive optimal error estimates we extend te ideas introduced in Section 2. We denote by a, te natural bilinear form associated wit A. We assume a.,. satisfies te following continuity and coercivity conditions. Assumption 1. Tere exist non-negative constants C a, C α, c a, c α suc tat 12

13 1. Continuity of te bilinear form and data: at; u, v c a u 2 + C a u c a v 2 + C a v and f, v f ca u 2 + C a u Corecivity of te bilinear form: at; u, u c α u 2 C α u 2 In tis context te natural weak formulation of 1.1 is to find u U L 2 [, T ; U] H 1 [, T ; U ] suc tat ut, vt + T u, v t + at, u, v = u, v + T F, v, 3.1 for all v U. We empasize tat U is te natural space to seek convergence, so ideally we would like to derive estimates in a related norm. To approximate te solution of te above weak formulation we introduce a partition = t < t 1 <... < t N = T of [, T ] and on eac partition we construct a closed subspace U n U. Te discontinuous Galerkin metod constructs an approximate solution u,t n ] P k, t n ; U n satisfying u n, v n + u, v t + a ; u, v = u n 1, v n tn F, v v P k, t n ; U n. Integration of te temporal term by parts yields te representation: 3.2 u t, v + a ; u, v + u n 1 + un 1, v+ n = F, v v P k, t n ; U n Preliminary Estimates. Classical bounds for te parabolic equation 3.2 are obtained upon selecting u = v in equation 3.1; te discrete analogue would be to set v = u in 3.3. Upon observing tat u t, u + u n 1 + un 1, u n 1 + = 1 2 un un un 1 + un 1 2, standard enery arguments use te continuity and coercivity assumptions 1 lead to te inequality u n 2 + c α u n u 2 + u n 1 u n ca /c α F 2 + 2C α + C a u 2. 13

14 Wen u P k, t n ; U n wit k 2 tis inequality does not control u s for s, t n. Wen u is piecewise constant or linear in time k = or 1 te terms on te left and side will dominate te last term on te rigt for sufficiently small time steps [24, Teorem 12.4]. Te case k > 2 as not been completly addressed previouly; typically strict coercivity is assumed so tat C α. In tis situation it is possible to write t m 1 F, u t m 1 1/2ɛ F 2 + ɛ/2 u 2 + u 2 and using te Poincare inequality to control te last term. Similar difficulties are encountered wit error estimates wen k 2. Letting e = u u te ortogonality condition becomes e n, v n + e, v t + a ; e, v = e n 1, v+ n 1 for all v P k, t n ; U n. Writing e = u u = u P u + P u u e + e p, were P is te projection defined in Definition 2.4 we compute e n, vn + e, v t + a ; e, v e n 1, v n 1 = e n p, v n + e p, v t + e n 1 p + tn, v+ n 1 a ; e p, v Since P u is te discontinuous Galerkin solution of an ordinary differential equation it follows tat e p = u P u satisfies te orgogonality condition 2.7. We ten conclude tat all but te last term on te rigt and side of te above expression vanises so tat e n, vn + e, v t + a ; e, v tn = e n 1, v n 1 + a ; e p, v. 3.5 Tis expresion is identical in form to te original sceme 3.2 for u wit F. = ae p,.. Putting v = e we obtain te analogue of equation 3.4, e n 2 + c α e n e 2 + e n 1 e n c a /c α c a e p 2 + C a e p 2 + 2C α + C a e 2. Again te natural energy arguments for te DG sceme fail to control e t for t, t n. 14

15 Remark 4. Te projection constructed using te discrete solution of te corresponding ODE is necessary wen different subspaces are used in eac time step, i.e., U n U n 1. Using te standard projection, P loc u in place of P as in [24] gives e n, vn + e, v t + a ; e, v tn = e n 1, v n 1 + a ; e n 1 p, v + e p, v n 1 +. wit e p = u P loc u. Note tat te last term is equal to en 1 p, v n 1 + w for every w U n 1, so wen U n = U n 1 we may pick w = v n 1 + to obtain Stability and Error Estimates. In tis section we sow ow te estimates in 3.4 and 3.6 can be augmented to provide bounds on te solution and te error for all times; in particular, for te intermediate times t, t n. To do tis we use te discrete caracteristic functions developed in Section 2.3. Since equations 3.2 for u and 3.5 for e are identical in form, te same line of argument can be applied to eiter of tem. For tis reason we will only prove te teorem for te error estimate and will simply state te analogous bound for u. Teorem 3.1. Let U H U be a dense embedding of Hilbert spaces and U be te subspace of L 2 [, T ; U] defined in Section 2.1 and let te bilinear form a : U U R and linear form F : U R satisfy Assumptions 1. Let u L 2 [, T ; U] H 1 [, T ; U ] be te solution of 3.1 and u U be te approximate solution computed using te discontinuous Galerkin sceme 3.2 on te partition = t < t 1 <... < t N = T and set τ max n t n. Ten tere exists a constant C > depending only on k troug te constant C k of Lemma 2.7, te constants C a, C α, and te ratio c a /c α, suc tat and n 1 1 λ u n 2 + λ sup us 2 + e Ctn 1 t i u i u i + 2 s t n +1 λ c α e Ctn s u s 2 ds T Oτ e Ctn u 2 + Cλ e Ctn s F s 2 ds, n 1 1 λ e n 2 + λ sup e s 2 + e Ctn 1 t i e i ei + 2 s t n i= +1 λ c α e Ctn s e s 2 ds T Oτ e Ctn e 2 + Cλ e Ctn s c a e p s 2 + C a e p s 2 ds, 15 i=

16 provided Cτ < 1. Here λ = 1/2C k + 4C k c a /c α + 1, 1, and e p = u P u and e = P u u were P : C[, T ; H] H 1 [, T ; U ] U is te projection defined in Definition 2.4. Proof. Since te line of argument to prove eac inequality is identical we only prove te second. Let ẽ P k, t n ; U n be te discrete approximation of χ [t n,te t constructed in Lemma 2.7. Setting v = ẽ in equation 3.5 and moving te term a.; e, ẽ to te rigt and side gives 1 2 e t en 1 e n = 1 2 en 1 2 a.; e p, ẽ + a ; e, ẽ. We estimate te eac of te last two terms seperatly. a.; e, ẽ c a e 2 + C a e 2 1/2 c a ẽ 2 + C a ẽ 2 1/2 t n 1 t n 1/2 c a e 2 + C a e 2 c a ẽ 2 + C a ẽ 2 t n 1 t n C k c a e 2 + C a e 2. Lemma 2.7 was used to bound ẽ in terms of e in te last line. A similar computation sows a.; e p, ẽ C k 2 Combining te above gives 1/2 c a /c α + 1c a e p 2 + C a e p 2 + c α e 2 + C a e 2. e t 2 + e n 1 e n e n C k c a /c α + 1c a e p 2 + C a e p 2 + c α 1 + 2c a /c α e 2 + 3C a e 2 We now consider te convex combination of 1 λ equation 3.4 and λ equation 3.7. We coose te coefficient, λ, so tat te term involving e 2 on te rigt of 3.7 is dominated by te corresponding term on te left of 3.4. Speciffically, let λc k 1 + 2c a /c α = 1/21 λ, or λ = Tis gives an estimate of te form 1 λ e n 2 + λ e t λ c α 2 e n Cλ 16 e 2 + e n 1 1 2C k + 4C k c a /c α + 1. e n c a e p 2 + C a e p 2 + e 2

17 were C depends only upon te coercivity constant c α and c a troug te ratio c a /c α. Bound te first and last terms on te rigt by and e n λ e n λ sup e s 2 t n 2 <s e 2 τ n sup e s 2, τ n t n, <s t n respectivly, and select te time t on te left so tat e t = sup <s t n e s to get 1 λ e n 2 + λ1 Cτ n sup e t λ c α e 2 + e n 1 <s t n 2 1 λ e n λ sup t n 2 <s e s 2 + Cλ e n c a e p 2 + C a e p 2. Upon introducing a factor 1 Cτ n in front of te first term, tis inequality takes te form 1 Cτ n α n + β n α n 1 + f n, and te teorem follows from te discrete Gronwall inequality. Remark 5. Note tat te above estimate consists of norms tat measure te beaviour of te solution at te nodal, interior and jump points. All norms included are te natural ones and no additional regularity is required on te rigt and side. Teorem 3.1 essentially sows tat te error for te parabolic equation can be bounded by te error of te ODE. If te norms.,. 2 and jump term J N e are defined by and v 2 = v 2 2 = T T sup vs 2 + c α e CT s vs 2 ds s T T e CT s vs 2 ds + c a e CT s vs 2 ds, J 2 Nv = N 1 i= e CT ti v i v i + 2, ten Teorem 3.1 states P u u 2 +JNP 2 u u CT P u u 2 + u P u 2 2 Since T ect s e p s 2 e CT sup s T e p s 2 various symetric error estimates follow. For example, setting u = P u and using te triangle inequlality gives u u 2 CT u P u 2, and u u CT u P u. Since P u is not a local projection, classical interpolation teory does not immediatly yield rates of convergence for a specific problem. However, we can use te results of. 17

18 Section 2.4 to estimate te rigt and sides of te above in terms of te local projection P loc u. Teorem 3.2. Under te assumptions in teorem 3.1 tere exists a positive constant CT depending only on k troug te constant C k of Lemma 2.7, te constants C, λ, C a, C α, and te ratio c a /c α, and te final time T suc tat te following estimate olds u u CT e + u Ploc u 2 + P u P loc u N CT e + u Ploc u 2 + P i I P i 1 ut i 1 + n c a max n P j I Pi 1 ut i 1 + P n... P 1 e, 1 n N j=i were e = P u u, and P loc u is te local projection defined in Definition 2.2, and P n : H U n is te ortogonal projection. Remark 6. 1 Upon assuming an inverse inequality of te form v C inv v for all v U n, n =, 1,..., and >, te above estimate simplifies to u u CT u P loc u c a C inv N P i I P i 1 ut i 1 + e. 2 Estimates for te jump terms J N u u can also be obtained; owever, tey will converge at a reduced rate; specifically, J N u u CT u P loc u 2 +J N u P loc u + N N P i I P i 1 ut i 1 + e + n c a max n P j I Pi 1 ut i 1 + P n... P 1 e, 1 n N j=i Appendix A. Discrete Gronwall Inequality. If 1 Cτ n a n + b n a n 1 + f n te discrete Gronwall inequality states tat if max n Cτ n < 1 ten a N + N n=1 b n N i=n 1 Cτ i a N N 1 Cτ i + f n N i=n 1 Cτ i. n=1 Since N exp Cτ i i=n 1 N i=n 1 Cτ i 1 1 N N Cτ i 2 exp Cτ i i=n i=n 18

19 and 1 N i=n Cτ i 2 1 C 2 T τ, were τ = max n τ n we may write a N + N e CtN t n b n 1 + T Oτ e CtN a + n=1 were t n = n τ n. N e CtN t n f n, n=1 REFERENCES [1] R. ADAMS, Sobolev Spaces, Academic Press, New York, [2] D.N. ARNOLD, F. BREZZI, B. COCKBURN AND D.L. MARINI, Unified analysis of discontinuous Galerkin metods for elliptic problems, SIAM J. Num. Anal. 39, 21-2, pp [3] M. BAUSE AND P. KNABNER, Uniform error analysis for Lagrange-Galerkin approximations of convection-dominated problems, SIAM J. Num. Anal. 39, pp [4] B. COCKBURN, G. E. KARNADIAKIS, AND C. W. SHU, Te development of discontinuous Galerkin metods, in Discontinuous Galerkin metods Newport, RI, 1999, Springer, Berlin, 2, pp 3-5. [5] M. DELFOUR, W. HAGER AND F. TROCHU, Discontinuous Galerkin Metods for Ordinary Differential Equations, Mat. Comp, 36, 1981, pp [6] J. DOUGLAS Jr. AND T. F. RUSSEL, Numerical metods for convection dominated difussion problems based on combining te metod of caracteristics wit finite element or finite difference procedures, SIAM J. Numer. Anal., 19, 1982, pp [7] T. DUPONT, Mes modification for evolutionary equations, Mat. Comp., 39, 1982, pp [8] T. F. DUPONT AND YINGJIE LIU, Symmetric error estimates for moving mes Galerkin metods for advection-diffusion equations, SIAM J. Numer. Anal. 4, 22, pp [9] K. ERIKSSON AND C. JOHNSON, Adaptive finite element metods for parabolic problems. I. A linear model problem, SIAM J. Numer. Anal., 28, 1991, pp [1] K. ERIKSSON AND C. JOHNSON, Adaptive finite element metods for parabolic problems. II. Optimal error estimates in L L 2 and L L, SIAM J. Numer. Anal., 32, 1995, pp [11] K. ERIKSSON, C. JOHNSON AND V. THOMÉE, Time discretization of parabolic problems by te discontinuous Galerkin metod, RAIRO Modél. Mat. Anal. Numér., 29, 1985, pp [12] D. ESTEP AND S. LARSSON, Te discontinuous Galerkin metod for semilinear parabolic equations, RAIRO Modél. Mat. Anal. Numér., 27, 1993, pp [13] P. JAMET, Galerkin-type approximations wic are discontinuous in time for parabolic equations in a varible domain, SIAM J. Numer. Anal. 15, 1978, pp [14] C. JOHNSON, Numerical Solution of Partial Differential Equations by te Finite Element Metod, Cambridge, [15] P. LASAINT AND P.-A. RAVIART, On a finite element metod for solving te neutron transport equation, in Matematical aspects of finite elements in partial differential equations, C. de Boor, ed., Academic Press, New York, 1974, pp [16] S. LARSSON, V. THOMÉE AND L.B. WALHBIN, Numerical solution of parabolic intregrodifferential equations by te discontinuous Galerkin metod, Mat. Comp. 67, 1998, pp

20 [17] YINGJIE LIU, R.E. BANK, T.F DUPONT, S. GARCIA AND R.P. SANTOS, Symmetric error estimates for moving mes mixed metods for advection-diffusion equations, SIAM J. Numer. Anal. 4, 23, pp [18] M. LUSKIN AND R. RANNACHER, On te smooting property of te Galerkin metod for parabolic equations, SIAM J. Numer. Anal. 19, 1982, pp [19] K. MILLER, Moving finite elements II, SIAM J. Numer. Anal. 18, 1981, pp [2] K. MILLER AND R.N. MILLER, Moving finite elements I, SIAM J. Numer. Anal. 18, 1981, pp [21] K.W. MORTON, A. PRIESTLEY AND E. SÜLI, Stability of te Lagrange-Galerkin metod wit nonexact integration, RAIRO Modél. Mat. Anal. Numér., 22, 1988, pp [22] SHOWALTER, R. E., Hilbert Space Metods for Partial Differential Equations, Pitman, [23] SHOWALTER, R. E., Monotone operators in Banac space and nonlinear partial differential equations, American Matematical Society, Providence, RI, [24] V. THOMÉE, Galerkin finite element metods for parabolic problems, Spinger-Verlag, Berlin, 1997 [25] N. J. WALKINGTON, Convergence of te discontinuous Galerkin metod for discontinuous solutions, preprint. 2