INTERIOR PENALTY DISCONTINUOUS GALERKIN METHODS WITH IMPLICIT TIME-INTEGRATION TECHNIQUES FOR NONLINEAR PARABOLIC EQUATIONS

Transcription

1 INTERIOR PENALTY DISCONTINUOUS GALERKIN METHODS WITH IMPLICIT TIME-INTEGRATION TECHNIQUES FOR NONLINEAR PARABOLIC EQUATIONS LUNJI SONG,, GUNG-MIN GIE 3, AND MING-CHENG SHIUE 4 Scool of Matematics and Statistics, Lanzou University, Lanzou 73, P.R. Cina Beiing Computational Science Researc Center, Beiing 84, P.R. Cina Department of Matematics, University of California Riverside, 9 University Ave., Riverside, CA 95, USA 3 Department of Applied Matematics, National Ciao Tung University, Taiwan 4 Abstract. We prove existence and numerical stability of numerical solutions of tree fully discrete interior penalty discontinuous Galerkin IPDG metods for solving nonlinear parabolic equations. Under some appropriate regularity conditions, we give te l H and l L error estimates of te fully discrete symmetric interior penalty discontinuous Galerkin SIPG sceme wit te implicit θ-scemes in time, wic include backward Euler and Crank-Nicolson finite difference approximations. Our estimates are optimal wit respect to te mes size. Te teoretical results are confirmed by some numerical experiments. key words. discontinuous Galerkin metod; error estimate; existence; numerical stability; nonlinear parabolic equation. Introduction Altoug te definition of discontinuous polynomial spaces emerged in te 97s [9, 8, 9], it was in te recent decade tat discontinuous Galerkin DG metods ave become attractive as a powerful simulation tool for solving partial differential equations see e.g. [5, 6,, 3, 5]. Te mixed DG and te primal DG are two main families of DG. Te local discontinuous Galerkin sceme LDG [4] is a representative of te mixed DG. Te primal DG metod depends on te appropriate coice of penalty terms for te discontinuous sape functions and as a different treatment of te diffusion term [7, 9, Correspondence to: Luni Song, Scool of Matematics and Statistics, Lanzou University, Lanzou 73, Cina song@lzu.edu.cn. Tis work was partly done wen tis autor worked as a visitor supported by Prof. Zimin Zang at Beiing Computational Science Researc Center CSRC. Contract grant sponsor: Young Scientists Fund of National Natural Science Foundation of Cina; contract grant number: 96 Contract grant sponsor: Tianyuan Fund for Matematics of National Natural Science Foundation of Cina; contract grant number: 665 Contract grant sponsor: Te Fundamental Researc Funds for te Central Universities; contract grant number: lzubky--65 Contract grant sponsor: National Science Council of Taiwan; contract grant number: -5- M-9-9-MY Contract grant sponsor: National Science Foundation of USA; contract grant number: DMS- 4.

2 L.J. SONG, G.M. GIE, AND M.C. SHIUE 6], wic can be referred to as Interior Penalty Discontinuous Galerkin IPDG metods. Tere are four primal DG versions, Symmetric Interior Penalty Galerkin SIPG [9, 5], Nonsymmetric Interior Penalty Galerkin NIPG [, 5], Incomplete Interior Penalty Galerkin IIPG [5] and Oden-Babuska-Baumann DG OBB-DG metods [6]. DG metods possess a few important features over oter types of finite element metods. For example, tey naturally andle inomogeneous boundary conditions and interelement continuity wit a weak enforcement; tey allow te use of nonuniform and unstructured meses and ave local conservation properties. In addition, tey appear to be non-oscillatory in te presence of ig gradients and roug solutions. But tey seem to introduce a relatively large number of degrees of freedom over inter-elements. Fortunately, two-level and multilevel preconditioners tend to remedy tis disadvantage [3, 8]. Te p version [] works efficiently and te accuracy is acieved if one geometrically refines te mes by grading towards te corners of te polygonal boundary were in general singularities of te exact solution occur, and if one appropriately cooses te polynomial degree p on eac subdivision. Arnold [] first analyzed a semidiscrete IPDG metod for solving nonlinear parabolic boundary value problems and stated optimal order error estimates in te energy and L norms. For reaction-diffusion equations, Georgoulis and Süli [9] presented fully p-optimal error bounds in te energy norm by introducing an augmented Sobolev space. Optimal convergence in L L for SIPG as also been establised for reactive transport in porous media, but sometimes NIPG and IIPG do not ave L L optimality [5]. For NIPG and IIPG, l L optimal error estimates were given for a tree-dimensional parabolic equation in a rectangular domain [7] and te L optimality was establised for polynomials of odd degrees for one-dimensional elliptic equations [4]. Rivière and Weeler [] derived a priori error estimates in te l H and l L norms for a fully discrete NIPG sceme wit a θ sceme for nonlinear parabolic equations. Recently, Om et al. [7] obtained an optimal l H and L L error estimates of a semi-discrete SIPG sceme for nonlinear parabolic equations, and generalized te l L error estimate to te backward Euler SIPG metod in [8]. But tere are no related numerical experiments presented. We consider te following nonlinear parabolic equation:...3 u t ax, u u = fx, u, in Ω, T, ax, u u n =, in Ω, T, u t= = ψx, on Ω }, were Ω is an open interval in R, or a convex polygonal domain in R, n is te unit outward normal vector to Ω, and T > is arbitrary but fixed. Te equation., supplemented wit te boundary condition. and te initial condition.3, describes te diffusion of a cemical species of te concentration u in a porous medium wit a source term fx, u. We assume tat a and f are uniformly Lipscitz continuous wit respect to te variable u, namely, tere exist positive constants L a and L f suc tat.4.5 ax, u ax, u L a u u, for u, u R, fx, u fx, u L f u u, for u, u R.

3 IMPLICIT IPDG METHODS FOR NONLINEAR PARABOLIC EQUATIONS 3 Moreover, we assume tat for any compact set S in R, tere exist positive constants K and K suc tat.6 < K ax, p K, < K p ax, p K, in Ω S. For te sake of simplicity, we also assume tat.7 f, =. In our study, u is assumed to be a strong solution. Tat is, u C Ω [, T], a solution of te problem.-.3, satisfies te regularity conditions below: u, ut L, T; H s Ω, for some s ;.8 u L Ω, T. In tis present work, te interior penalty discontinuous Galerkin scemes for approximating solutions of te problem.-.3 are analyzed, and for a fully discrete θ sceme in time te optimal l L error estimates are derived under some appropriate regularity conditions. Our analysis focuses on te following issues wic ave not yet been adequately considered in te literature: Based on an implicit θ sceme, existence of solutions of te fully discrete IPDG scemes will be proven. Wen implicit θ time-integration tecniques are considered in order to avoid rigid CFL stability conditions associated wit te mes size, numerical stability of tese fully discrete IPDG scemes will be analyzed. In te p version, optimal l H and l L error estimates will be given for te fully discrete SIPG sceme wit implicit time-integration scemes. Some numerical results are presented in our work. As we know, few numerical results from an implicit time-stepping IPDG sceme ave been publised for solving nonlinear parabolic equations. Te content of tis article is summarized as follows. In Sect. we recall a few definitions and te formulation of te interior penalty DG formulations in a semi-discrete form and in fully discrete forms wit a θ sceme. For te semi-discrete DG sceme, we state some properties for IPDG scemes in Sect. 3, and derive existence and numerical stability for te fully discrete IPDG scemes in Sect. 4 and Sect. 5, respectively. In Sect. 6 we give a unified analysis of optimal a priori error estimates in te l H and l L norms for te fully-discrete SIPG sceme wit implicit θ time-integration tecniques. Finally, te numerical results are presented to sow te effectiveness of te DG metods in Sect. 7.. Te discontinuous Galerkin metod We subdivide te domain Ω into elements E, E,, E N, were E i is an interval in D or a triangle in D and N is te number of all elements. Here > denotes te maximal diameter of all elements. For eac >, we write te resulting subdivision in te form: E := E, E,, E N }.

4 4 L.J. SONG, G.M. GIE, AND M.C. SHIUE Note tat Ω = N i= Ēi, for eac >. We set, for eac element of E, i = te diameter of E i, i N,. ρ i = te radius of te largest ball inscribed in E i, i N. We impose a regularity assumption on te mes E, i.e., tere exists a constant ζ >, independent of, suc tat min ρ i ζ. Moreover, we assume tat te mes E is quasi-uniform: tere exists a constant τ >, independent of, suc tat. min i τ. We introduce te set F of edges of te mes E : F := e, e,, e P, e P +,, e M }, were e i Ω, if i P, e i Ω, if P + i M. On eac e i i M of E, we fix a unit outer normal vector n i : te unit normal vector on e i, pointing from E k to E, if e i = E k E, n i = te unit normal vector on e i, pointing outward of Ω, if P + i M, ten we denote te average and ump operators below: For v H s E, s >, v} ei := v E ei + v E k ei, if e i = E E k, i P, v Ek ei, if e i = E k Ω, P + i M. v Ek ei v E ei, if e i = E E k, i P, [v] ei := v Ek ei, if e i = E k Ω, P + i M. For brevity, we drop te subscript e i of tese two operators trougout tis paper. Along tis article, te H m Sobolev norm on ω is defined by m,ω for a positive integer m, i.e.,.3 m,ω := H m ω, m <, ω R or R. Note tat, by default, H ω denotes L ω wit te L inner product, and, ω is te standard L -norm on ω. Ten using.3, we introduce te broken Sobolev space for any real number s: H s E = v L Ω : v Ei H s E i, i N }, wic is equipped wit te broken space norm: v s := v H s E = N /. v s,e i i=

5 IMPLICIT IPDG METHODS FOR NONLINEAR PARABOLIC EQUATIONS 5 Also, given a time interval [a, b], we use te broken Sobolev L H s and L H s norms: b v L a,b;h s = v, t s dt, We also introduce a space of test functions.4 were, for eac i N, It is clear tat a v L a,b;h s = ess sup v, t s. t a,b D r E = v L Ω : v Ei P r E i, i N }, P r E i := te space of polynomials of total degree at most r on E i }. D r E H E L Ω. However, since te test functions in D r E are discontinuous along te edges e i, i P in Ω, we notice tat D r E H Ω. We tus introduce te interior penalty term J σ : D r E D r E R in te form: P J σ v, w = σ k.5 [v][w]ds, e k e k k= wic penalizes te ump of te functions across te edges e k, k P. Here te penalty parameter σ k is a nonnegative real number to be cosen and e k is te Lebesgue measure of te edge e k. It is easy to see tat.6 e i i, i N. We also define te energy norm on D r E trougout tis paper:.7 v DG = N /, v,e k + J σ v, v v Dr E. k= Aiming to study te strong solution u C Ω [, T], satisfying te regularity conditions.8, of te problem.-.3, we proceed element by element as appears in [3]. As a result, we obtain te following consistent weak formulation of te problem.-.3: Find ut H s E, s > 3, suc tat.8.9 u t, v + A ǫ u; u, v = fu, v, v H s E, u, v = ψ, v, were A ǫ ρ; v, w is bilinear in te last two terms: N P A ǫ ρ; v, w = aρ v wdx aρ v n k }[w]ds = E k= e k. P + ǫ aρ w n k }[v]ds + J σ v, w, v, w Hs E. e k k= Here, te parameter ǫ in A ǫ may take te value, or ; see Remark..

6 6 L.J. SONG, G.M. GIE, AND M.C. SHIUE To discretize te problem.8, we first introduce te following notations. For any smoot function φ : Ω [, T] R, we set. φ n := φx, t n, t n = n n N, were N is a positive integer and = T/N. Ten for an arbitrary but fixed θ, we set. φ n θ = θ φn + + θ φn+, n N. To discrete.8, by.-., we use te discontinuous Galerkin metod for te spatial variable, and te θ sceme for te time variable. Now te fully discrete IPDG formulation is to seek a sequence u n }N n of functions in D re suc tat n,.3 u n+ u n, v + A ǫ u n,θ; u n,θ, v = fu n,θ, v, v D r E,.4 u = ũ, were ũ is a L proection of ψ onto D r E and u n,θ = θ un + +θ un+. Remark.. Note tat for a fixed function ρ, A ǫ appearing in. is symmetric if ǫ =, and is non-symmetric if ǫ = or. Moreover, depending on te value of ǫ, te discontinuous Galerkin metod considered in.3 is referred to SIPG if ǫ = ; NIPG if ǫ = ; or IIPG if ǫ =. For te coice of penalty parameters σ k of tese discontinuous formulations, see Georgoulis and Süli [9], Rivière et al. [], and te references terein. Remark.. If θ =,.3 yields te Crank-Nicolson discontinuous Galerkin approximation; If θ =,.3 becomes te backward Euler discontinuous Galerkin approximation. 3. Some estimates of te IPDG scemes In tis section, we mainly state some approximation results in te space of polynomials of degree r, wic will be used later. And we denote by C a generic positive constant. Lemma 3.. Assume tat u H s Ω, for s and let r and assume tat ā is a given positive constant. Ten tere exists an interpolant û D r E of u satisfying tat see [] 3. ā û u n k }ds =, k =,, P, e k û u,e C µ r s u s,e, E E, û u,e C µ r s u s,e, E E, û u,e C µ r s u s,e, E E, û u,e C µ r u s s,e, E E, û,ek C u,ei E, for e k = E i E, were µ = minr +, s and C is independent of, s and r.

7 IMPLICIT IPDG METHODS FOR NONLINEAR PARABOLIC EQUATIONS 7 Lemma 3.. For eac E k E, let e k be an edge of E k and let n k be a unit vector normal to e k. Ten tere exists a positive constant C depending only on τ and r defined in. and.4 suc tat te following two trace inequalities are valid []: v,e k C E k v,e k + Ek v,e k, v H E k, v n k C E k v,e k + Ek v,e k, v H E k.,e k Lemma 3.3. For eac E k E and v P r E k, let e k be an edge of E k and let n k be a unit vector normal to e k. Ten tere exists a positive constant C depending only on r and τ suc tat te following two local inverse inequalities old []: 3.9 v,ek C E k v,ek, r, v n k C E k v,ek.,ek Squaring te second inequality 3.9, multiplying it by e k and summing for k = to P, we obtain Lemma 3.4. Tere exists a positive constant C Ω,τ depending on Ω and τ suc tat P } e k v 3. C Ω,τ v n, v H E. k,e k k= We define a new bilinear form: wit a positive real number λ. A ǫ,λ ρ; v, w = A ǫ ρ; v, w + λv, w, In [4], we note tat A ǫ is coercive. If te positive constant δ is suc tat k= K C Ω,τ > δ > ǫ K 4 min k σ k } and te penalty parameters σ ǫ K k satisfy minσ k } > C Ω,τ, ten for k K any v, ρ D r E, P A ǫ ρ; v, v α v + } e k v 3. + J σ v, v, n k,e k K were α = min, K C Ω,τ δ, ǫ K 4δ minσ k } }. Tus, if we consider an NIPG metod ǫ =, ten we can coose σ k > and α = mink, }. For te SIPG and IIPG metods, te penalty parameters σ k will be cosen sufficiently large, as te ratio K K or K δ becomes extremely large. Based on te estimate 3. and te definition of A ǫ,λ, Ten we ave te following lemma: K Lemma 3.5. Assume tat C Ω,τ > δ > ǫ K and min ǫ 4minσ k } kσ k } > C K Ω,τ K, ten tere exists a positive constant κ independent of and r suc tat 3. A ǫ,λ ρ; v, v κ v DG, ρ, v D re. Here one can coose κ = α. Te following lemma can be proven similarly as in [4].

8 8 L.J. SONG, G.M. GIE, AND M.C. SHIUE Lemma 3.6. Tere exists a positive constant C independent of and r suc tat 3.3 A ǫ,λ ρ; v, w C v DG w DG, ρ, v, w D r E. By Lemma 3.6, we notice tat A ǫ is continuous: tere exists a constant β > suc tat 3.4 A ǫ ρ; v, u β v DG u DG, ρ, v, u D r E. Te Aubin-Nitsce lift tecnique is well suited to te analysis of te DG metod for linear problems, since te SIPG sceme is symmetric. But for te nonlinear parabolic equation, we will use te following proection and lemma as in [7]. Let u H Ω. Te Galerkin proection π u D r E of u is defined by requiring tat 3.5 A,λ u; u, v = A,λ u; π u, v, v D r E. π u is a function mapping from, T onto D r E and its unique existence follows by te Lax-Milgram Teorem. We denote and state te following estimates. π u t = t π u, Lemma 3.7. For te SIPG sceme ǫ = and r, s, tere exists a constant C satisfying were µ = minr +, s. u π u DG C µ r s u s, u π u C µ r s u s, u t π u t DG C µ r s u s + u t s, u t π u t C µ r s u s + u t s, A straigtforward modification of te analysis of Teorem 4. in [7] yields te proof of Lemma 3.7, and we omit te proof. 4. Existence of a fully discrete solution Te existence of te fully discrete IPDG formulation.3 is to find a sequence u n }N n= of functions in D r E. We need te following Lemma in [6] to sow te existence of a fully discrete solution u n+ in.3. Lemma 4.. Let X be a finite dimensional Hilbert space wit scalar product, and norm and let P be a continuous mapping from X into itself suc tat 4. Pξ, ξ >, for ξ = K >, ten tere exists ξ X, ξ K suc tat P ξ =.

9 IMPLICIT IPDG METHODS FOR NONLINEAR PARABOLIC EQUATIONS 9 Let M be te dimension of D r E corresponding to te number of degrees of freedom DOF. We coose a basis φ i } M i= of D re made of polynomials φ E i wit supp φ E i E, E E, and degree of φ E i less tan r, for i M, N. Ten, tanks to Lemma 4., we state and prove te following existence result of a discrete solution u n, n N, of.3. Teorem 4.. Under te assumptions of Lemma 3.5, and if 4. < L f, ten tere exists a solution u n+ D r E, n N of te parabolic equation.3 in weak form. Proof. We first observe tat u D re by te definition of u in.4. To complete te proof, we assume tat tere exists u n D re, n l, solution of te IPDG formulation.3. Ten, to sow te existence of u n+ D r E, we consider.3 wit n = l. Aiming to find u l+ = M i= ξ iφ i, we set a mapping 4.3 were Pξ = P ξ, P ξ,, P M ξ, ξ = ξ, ξ,, ξ M T, u l+ u l P i ξ :=, φ i + A ǫ u l,θ ; ul,θ, φ i fu l,θ, φ i, i M. We apply Lemma 4. wit X = R M wic is equipped wit te inner product, and te l norm l. Ten we find M Pξ, ξ = P i ξξ i i= u l+ u l = u l+ u l =, u l+ + A ǫ u l,θ ; ul,θ, ul+ fu l,θ θ, u l+ f ul + + θ ul+ + A ǫ θ ul + + θ ul+ ; θ ul + + θ, ul+, u l+ ul+, ul+ Assume tat u l DG is finite. Te operator P is continuous and tere remains to ceck 4.; for tis purpose, we consider te scalar product Pξ, ξ. We ave u l+ u l, u l+ = ul+ ul + ul+ u l, and θ A ǫ ul + + θ ul+ ; θ ul + + θ = θ θ A ǫ ul + + θ ul+ ; ul, ul+ ul+, ul+.

10 L.J. SONG, G.M. GIE, AND M.C. SHIUE + + θ θ A ǫ ul + + θ ul+ ; ul+, ul+ θ β u l DG u l+ DG + + θ α u l+ DG + θ α u l+ DG θ 4 α u l+ DG θ β 4α ul DG + 3θ 4 α u l+ DG θ β 4α u l DG, were te first inequality is derived by te coercivity 3. and te continuity 3.4 and te second inequality olds by te Caucy-Scwarz inequality. We bound te tird term θ f ul + + θ ul+, u l+ u l = f + u l+ + θ ul+ u l, u l+ L f ul + ul+ + θ ul+ u l u l+ L f L f ul+ ul+ L f u l+ + L f + L f 8 ul + ul+ + θu l ul+ + L f 4 ul + ul+ + θ L f ul + θ L f 4 ul+ 4 ul+ u l u l L f u l+ + L f C ul DG + θ L f 4 ul+ u l. Te last inequality is derived by a generalization of Poincaré inequality to te broken Sobolev space H E see [3]. Ten, we ave Pξ, ξ L f u l θ 4 α u l+ DG Terefore, using te assumption 4. gives 4.4 θ L f β θ + C L f 4α Pξ, ξ + 3θ β α u l+ 4 DG θ + C L f 4α + C u l+ u l u l DG. + C u l DG. We need to find a suitable ξ R M to make te rigt and side of 4.4 positive. To do tis, since u l+ = M i= ξ iφ i, we set and write ξ = ξ,,,, R M, u l+ DG = ξ φ DG = ξ φ DG. Ten we coose ξ = K > large enoug so tat + 3θ β α K θ + C L f 4 4α + C u l DG >.

11 IMPLICIT IPDG METHODS FOR NONLINEAR PARABOLIC EQUATIONS Tis sows tat tere exists a vector ξ R M wit ξ l = K > suc tat Pξ, ξ >. By Lemma 4., tis implies te existence of u l+ D r E, wic completes te proof. 5. Numerical stability of te fully discrete IPDG scemes Now we prove a new stability result for tese fully discrete IPDG scemes Teorem 5.. Under te same assumptions of Lemma 3.5, and let te source term f satisfy te conditions.5 and.7. If a time step satisfies tat 5. < L f, ten numerical solution of te fully discrete problem.3-.4 is stable in te following sense: u n e4l ft u, n N, N n= Proof. Taking v = u n,θ u n+ 5.4 u n We recall te following two identities: u n,θ DG e 4LfT u α, θ [, ]. in.3, we get, u n,θ + A ǫ u n,θ; u n,θ, u n,θ = fu n,θ, u n,θ. a b, a = a b + a b, a b, b = a b a b. Using te two identities, it is easily proved tat u n+ u n, u n,θ = 5.5 u n+ un + θ un+ u n. Using 5.5, 3. and te Caucy-Scwarz inequality, we infer from 5.4 tat 5.6 u n+ un + θ un+ u n + α u n,θ DG fun,θ u n,θ. On te oter and, using.5 and.7, we estimate te rigt and side of fu n,θ u n,θ L f u n,θ L f + θ un+ + θ un L f 4 un+ + u n + θun+ u n L f un+ + u n + θ L f u n+ u n L f u n+ + L f u n + θ L f u n+ u n.

12 L.J. SONG, G.M. GIE, AND M.C. SHIUE Combining 5.6 and 5.7, after multiplying by on bot sides, we find 5.8 L f u n+ + θ θl f u n+ u n + α u n,θ DG + L f u n, Due to te assumption tat < L f, te above inequality 5.8 can be rewritten in te form: 5.9 were u n+ + A un+ u n + B un,θ DG D un, A = θ θl f L f >, B = α L f >, D = + L f L f >. Multiplying 5.9 by D n and summing on n from n = to m for all m N, we get 5. D m u m + m n= Since D >, 5. gives tat m AD n u n+ u n + n= BD n u n,θ DG D u. 5. m u m + n= A u n+ m u n + n= B u n,θ DG DN u, m N. Due to te fact tat +v v ev, < v <, we observe tat 5. D N u e4l f N u e4l ft u. Combining 5. and 5., we find tat 5.3 u m e4l ft u, m N, m 5.4 B u n,θ DG e4l ft u, m N. n= Te above second inequality follows N n= wic concludes te teorem. u n,θ DG α L f e 4L ft u α e 4L ft u, Remark 5.. Under te same condition as tat in te existence Teorem 4., we ave proved numerical stability for te fully discrete implicit IPDG metods. Also, te analogous proof can be given for a fully discrete explicit IPDG sceme. If te source term f = fx, u is locally Lipscitz continuous in its argument u as in [5], one can give a similar numerical stability.

13 IMPLICIT IPDG METHODS FOR NONLINEAR PARABOLIC EQUATIONS 3 6. Error estimates of te fully discrete SIPG sceme In tis section, we restrict our attention to te SIPG case. Coosing ǫ = in.3 and no constraints on grid sizes and time steps required, we will sow an error estimate of te implicit time stepping SIPG metod, wile te time derivative is discretized in time by te θ sceme. Define te fully discrete l L and l H norms 6. v l L = max =,,N v, v l H = N = v /. Using te notation., we set 6. t θ = θ t + + θ t +, θ, N. Ten, we first give te following lemmas. Lemma 6.. For a sufficiently regular u = ux, t, we consider π u D r E were π u = π ut, N. Ten we ave 6.3 π u + π u = π u t t θ + ρ,θ, N, x Ω, were for t t θ, t +, t t, t θ, and for s, ρ,θ = θ + θ π u tt t θ + θ 3π u ttt t θ 3π u ttt t, 6 In te case θ =, we also ave ρ,θ C u tt L t,t + ;H s. ρ,θ C u ttt L t,t + ;H s, s, were C and C are two constants independent of u, π u, and. Proof. By applying te Taylor expansion to π u at t = t θ, 6.3 easily follows. More results can be found similarly in []. Lemma 6.. For any edge e k of F and any element E k of E, π u in te L norm is bounded by a positive constant C depending on u and independent of,, r, and s, i.e., π ut,ek < C, π ut,ek < C, t [, T], t [, T].

14 4 L.J. SONG, G.M. GIE, AND M.C. SHIUE Proof. We consider a given edge e k = E m E n and E mn = E m E n, ten we ave by using Lemmas π u,ek û,ek + π u û,ek C u,emn + C π u û,ek C u,emn + C π u û,emn + π u û,emn 6.6 C u,emn + C π u û,emn C u,emn + c u π u,emn + u û,emn C u,emn + c u π u + u û C u,emn + C µ r s u s <, wic leads to 6.4. For u H Ω, one can bound π u,ek π u u,ek + u,ek CmeasE k π u u,ek + u,ek C µ r s u s + u < C. Consequently, 6.5 is proven. We now state a priori l L and l H error estimates, wic is optimal, for te finite element system.3 given by te fully discrete implicit time-stepping discontinuous Galerkin metod. Note tat our result below is a generation of Teorem 4. in [7], wic is not involved in any time discretization. However, as appears in A, te main idea in [, 7] can be adapted in our current situation. Teorem 6.. Let te solution u of te nonlinear parabolic equation.-.3 satisfy te regularity properties u L Ω, T, and u, u t L, T; H s Ω, s. In addition, we assume tat for θ, ], and in te case θ =, u t L, T; H s Ω, 3 u t 3 L, T; H s Ω. Moreover, we assume tat te initial condition u x, = ũ D r E satisfies 6.7 u π u C µ r s ψ s, µ = minr +, s.

15 IMPLICIT IPDG METHODS FOR NONLINEAR PARABOLIC EQUATIONS 5 Under te same assumptions of Teorem 4., ten u }N = te numerical solution of te fully-discrete time SIPG sceme ǫ = in.3 satisfies 6.8 and u u l L + u u l H N C µ ψ r s 4 s + ut s + u t t s N + = = u tt L t,t + ;H s, < θ, 6.9 u u l L + u u l H C µ r s 4 N + 3 ψ s + N = = ut s + u tt s u ttt L t,t + ;H s, θ =, were µ = minr +, s, r and C is depending on u and independent of, r and. Remark 6.. We conclude tat in te case < θ, te error is O µ +, so as r s 4 to get an optimal convergence order, we need te restriction = O µ. Te similar r s 4 restriction 3 = O µ appears to te oter case θ =. Due to µ >, so we can r s 4 coose te case C numerically. Remark 6.. For te fully implicit SIPG scemes, in addition to te regularity setting.8, Teorem 6. requires te solution u satisfying u tt L, T; H s Ω and u ttt L, T; H s Ω corresponding to te cases of < θ and θ =, respectively, to assure te temporal accuracy. Compared to te fully implicit SIPG scemes, in te interior of eac element, te fully explicit SIPG sceme in [4] only requires a less smoot solution satisfying u L, T; H s Ω, u t L, T; H s Ω and u tt L, T; H Ω. It appears tat te fully implicit scemes need more regularity assumptions in time tan te fully explicit one. 7. Numerical results In tis section we present some numerical experiments to illustrate te performance of our fully discrete SIPG metods for nonlinear parabolic equations. We will also sow te performance of Euler backward and Crank-Nicolson scemes in te time discretization. We consider te following nonlinear equation on te domain Ω =, 7. u t u u = fx, u, in Ω, T. Te exact solution is given by u = +cosπx cosπyexp t. Te initial and boundary conditions and te rigt-and side function f can be obtained by using te exact solution. Te final time T = is taken and te penalty parameter σ k is uniformly defined by σ k = r + on eac edge.

16 6 L.J. SONG, G.M. GIE, AND M.C. SHIUE Te error of te solution is evaluated over te domain Ω in te L, T; H Ω seminorm error L H = u u l H. We use te SIPG metod in various quasi-uniform meses for spatial discretization respectively, and te backward Euler and Crank-Nicolson metods for time integration wit a uniform time step. We compute te errors corresponding to te exact solution ux; t in te l L and L, T; H Ω norms. Tables -3 demonstrate te errors and order of convergence of te fully discrete scemes, wic confirm optimality as sown in Teorem 6.. Table 3 sows tat convergence order of te numerical solution decreases in te l L norm and almost keeps invariant in te L, T; H Ω norm, wile θ increases from to. Figure illustrates tat te isolines of te numerical solutions do not ave spurious oversoots as te polynomial degree increases. Table. Convergence order of te SIPG metod.3 wit Crank-Nicolson discretization in time. Here te mes size is =.88 and te time step is =.. error l L error L H order order r = 5.78E.E E.3E.68 r = 3.43E 3.9E E.3E.98 r = 3 3.4E 4.53E E.3E r = E 5.8E E 3.78E Table. Convergence order of te SIPG metod.3 wit Euler backward discretization in time. Here te mes size is =.88 and te time step is =.. error l L error L H order order r = 5.79E.E E.3E.68 r = 3.45E 3.E E.3E.98 r = E 4 6.9E E.3E r = 4 7.E 5 3.3E E 3.76E To weaken te condition of au, we consider anoter analytic solution u = sinπx sinπy exp t, wic satisfies te Neumann boundary condition, but does not satisfy te condition au > on te boundary. From Tables 4-6, we observe tat tese IPDG scemes are convergent and confirm te effectiveness of te fully discrete scemes. Table 6 sows tat convergence order of te numerical solution decreases in te l L norm and almost keeps invariant in te L, T; H Ω norm, wile θ increases from to. In Figure, te approximate solutions do not ave spurious oversoots in P element and appear more smoot on te boundary wit increasing polynomial degree r.

17 IMPLICIT IPDG METHODS FOR NONLINEAR PARABOLIC EQUATIONS 7 Table 3. Convergence order of te SIPG metod.3 wit θ scemes in time for second order polynomial approximation r =. Here te mes size is =.88 and te time step is =.. error l L error L H order order θ = 3.4E 3.9E E.3E.98 θ =. 3.43E 3.E E.3E.98 θ =. 3.45E 3.4E E.3E.98 θ = E 3 3.E E.3E.98 θ =.4 3.5E E E.3E.98 θ =.5 3.5E E E.3E.98 θ = E 3 5.9E E.3E.99 θ = E E E.3E.99 θ =.8 3.6E E E.3E.99 θ = E E E.3E.99 θ = 3.67E 3 8.E E.3E Figure. Isolines of te numerical solutions for P top left, P top rigt, P 3 bottom left and P 4 bottom rigt elements on te mes, SIPG formulation wit Crank-Nicolson time discretization.

18 8 L.J. SONG, G.M. GIE, AND M.C. SHIUE Table 4. Convergence order of te SIPG metod.3 wit Crank-Nicolson discretization in time. Here te mes size is =.88 and te time step is =.. error l L error L H order order r = 4.55E.3E E.3E.39 r = 6.9E 3 6.5E E.4E.36 r = 3.56E 3.77E E.E.7 r = E 4.3E E.6E 3 4. Table 5. Convergence order of te SIPG metod.3 wit Euler backward discretization in time. Here te mes size is =.88 and te time step is =.. error l L error L H order order r = 4.55E.3E E.3E.39 r = 6.86E E E.43E.35 r = 3.56E 3.88E E.3E.68 r = E 4.76E E.6E 3 4. Table 6. Convergence order of te SIPG metod.3 wit θ scemes in time for second order polynomial approximation r =. Here te mes size is =.88 and te time step is =.. error l L error L H order order θ = 6.7E E E.37E.35 θ =. 6.67E E E.38E.35 θ =. 6.6E E E.38E.35 θ = E 3 6.9E E.39E.35 θ = E E E.39E.35 θ = E E 4.86.E.4E.35 θ = E 3.3E E.4E.35 θ =.7 6.4E 3.7E E.4E.35 θ = E 3.3E 3.8.3E.4E.35 θ = E 3.46E 3..3E.4E.35 θ = 6.3E 3.6E E.43E Conclusions We ave analyzed te fully discrete IPDG metod wit a class of implicit θ scemes in time for a class of nonlinear parabolic equations in D or D. Te existence of numerical solutions and te numerical stability of tese fully discrete IPDG scemes are proven and tey ave te same restricted condition in time step. It is interesting tat te stability condition is not associated wit te mes size. Using a nonlinear elliptic proection

19 IMPLICIT IPDG METHODS FOR NONLINEAR PARABOLIC EQUATIONS Figure. Isolines of te numerical solutions for P top left, P top rigt, P 3 bottom left and P 4 bottom rigt elements on te mes, SIPG formulation wit Crank-Nicolson time discretization. and te interpolant approximation for te SIPG sceme, we ave proved a priori error estimates in te discrete l H semi-norm and in te l L norm, wic are optimal in. Te numerical results ave confirmed te presented teory. Te approximation spaces are considered on a quasi-uniform triangular mes, but our results can be extended to quadrilateral meses. Analogously, error estimates and stability analysis can also be carried out in 3D. Appendix A. Proof of Teorem 6. Proof. To tis end, we proceed in te following steps. Step representation of an identity for π u u. For n =, we write.3 in an equivalent form A. u + u, v + A,λ u,θ ; u,θ, v = fu,θ, v + λu,θ, v, v D re.

20 L.J. SONG, G.M. GIE, AND M.C. SHIUE At t = t θ, defined by 6., we notice tat te exact solution ut θ satisfies tat v D r E, A. ut θ t, v + A,λ ut θ ; ut θ, v = fut θ, v + λut θ, v. Subtracting A. from A., we find A.3 ut θ t u +, v + A,λ ut θ ; utθ, v A,λu,θ ; u,θ, v u, v = fut θ fu,θ, v + λut θ u,θ, v, v D re. By taking û D r E as an interpolant of u, wic satisfies Lemma 3., we define η θ = ut θ π ut θ, β θ = π ut θ û θ, ξ θ = π ut θ u,θ. We also write ξ = π u u. Ten, it follows from A.3 A.4 ξ + ξ = π u + π u, v + A,λ u,θ ; ξ θ, v u t t θ, v + A,λu,θ ; π ut θ, v A,λut θ ; ut θ, v + fut θ fu,θ, v + λut θ u,θ, v. Tanks to Lemma 6., it is easy to see tat A.5 π u + π u u t t θ = π u t t θ u tt θ + ρ,θ = η t θ + ρ,θ. Moreover, using 3.5 and te identity tat A,λ ut θ ; η θ, v =, we get A.6 A,λ u,θ ; π ut θ, v A,λut θ ; ut θ, v N = au,θ aut θ π ut θ v dx k= E k P au,θ aut θ π ut θ n k} [v] ds k= e k P au,θ aut θ v n k} [π ut θ ] ds. e k k=

21 IMPLICIT IPDG METHODS FOR NONLINEAR PARABOLIC EQUATIONS Substituting A.5 and A.6 into A.4, and taking v = ξ θ, we obtain A.7 ξ + ξ, ξ θ + A,λ u,θ ; ξ θ, ξ θ = η t θ, N ξ θ + au,θ aut θ π ut θ ξ θ dx k= E k P au,θ aut θ π ut θ n } k [ξ θ ]ds k= e k P au,θ aut θ ξ θ n k} [π ut θ ]ds e k k= + fut θ fu,θ, ξ θ + λut θ u,θ, ξ θ + ρ,θ, ξ θ I + I + I 3 + I 4 + I 5 + I 6 + I 7. Step estimates on I i, i 7. Now we derive te bounds for eac I i, i 7. We bound te first term I by te Caucy-Scwarz inequality A.8 I = η t θ, ξ θ ηt θ ξ θ C η t θ + ξ θ. Using 6.5, we estimate te term I : A.9 N I C u,θ ut θ π ut θ ξ θ dx E k k= C π ut θ N,E k= C η θ + ξ θ + α ξ θ. η θ,e k + ξ θ,e k ξ θ,e k To deal wit I 3, we consider a given edge e k = E m E n and E mn = E m E n. Ten using.4 and 6.4, we first notice tat e k au,θ aut θ π ut θ n k } [ξ θ ]ds A. C π ut θ,e k ut θ u,θ },e k [ξ θ ],e k C η θ },e k + ξ θ },e k [ξ θ ],e k α σ k e k [ξ θ ],e k + C η θ,e mn + η θ,e mn + ξ θ,e mn,

22 L.J. SONG, G.M. GIE, AND M.C. SHIUE and ence, a bound of te tird term I 3 follows: P I 3 = au,θ aut θ π ut θ n } k [ξ θ ]ds e k A. k= To estimate I 4, we observe tat N αj σ ξ θ, ξ θ + C η θ,e mn + η θ,e mn + ξ θ,e mn k= α ξ θ DG + C η θ + η θ + ξ θ. e k au,θ aut θ ξ θ n k } [η θ ]ds A. C ξ θ,e k u,θ ut θ },e k [η θ ],e k C ξ θ,e k η θ },e k + ξ θ },e k η θ,e mn + η θ,e mn C ξ θ,e k η θ,e mn + η θ,e mn + ξ θ,e mn η θ,e mn + η θ,e mn C 3 ξ θ,e k η θ,e mn + η θ,e mn + ξ θ,e mn η θ + η θ C ξ θ,e mn η θ,e mn + η θ,e mn + ξ θ,e mn ut θ C ξ θ,e mn η θ,e mn + η θ,e mn + ξ θ,e mn, and tus, we get te bound of te fourt term I 4 P I 4 = au,θ A.3 aut θ ξ θ n } k [η θ ]ds e k k= Te terms I 5, I 6, I 7 are easy to estimate: A.4 α ξ θ DG + C η θ + η θ + ξ θ. I 5 = fut θ fu,θ, ξ θ C ut θ u,θ ξ θ C η θ + ξ θ. A.5 I 6 = λut θ u,θ, ξ θ C ut θ u,θ ξ θ C η θ + ξ θ. A.6 I 7 = ρ,θ, ξ θ C ρ,θ + ξ θ. Step 3 summing up. To make te estimates on I i, i 7 useful, we first notice te following inequality olds true:. ξ ξ + ξ + ξ, ξ θ

23 IMPLICIT IPDG METHODS FOR NONLINEAR PARABOLIC EQUATIONS 3 Ten, using te bounds of I, I,, I 7 and te coercivity 3. of A,λ, we obtain from A.7 A.7 ξ + ξ + κ ξ θ DG C η t θ + ξ θ + η θ + η θ + ρ,θ + α ξ θ + ξ θ + ξ θ DG. By using te definition of DG, we infer from A.7 tat A.8 ξ + ξ κ + ξ θ DG C η t θ + ξ θ + η θ + η θ + ρ,θ, for a sufficiently small α. Multiplying A.8 by and summing up for =,, m, m N, we find A.9 m ξ m ξ + κ ξ θ C = m m ξ + C ηt θ + η + η = m + C 3 = = ρ,θ, m N. By discrete Gronwall s lemma, we find tat, for sufficiently small, A. m ξ m + C = m C ξ + C m + C 3 = = ρ,θ. ξ θ ηt θ + η + η Under te assumption 6.7, we obtain, for θ, ], tat A. m ξ m + C = ξ θ C µ r s 4 ψ s + C µ r m ut s 4 s + u t t s = m + C u tt L t,t + ;H, m N, s =

24 4 L.J. SONG, G.M. GIE, AND M.C. SHIUE and, in te case θ =, A. m ξ m + C = ξ θ C µ r s 4 ψ s + C µ r m ut s 4 s + u t t s = m + C 3 u ttt L t,t + ;H s, m N. = Inequalities A. and A. conclude te proof wit te use of Lemma 3.7. References [] D.N. Arnold, An interior penalty finite element metod wit discontinuous elements, SIAM J Numer Anal 9 98, [] C.E. Baumann, J.T. Oden, A discontinuous p finite element metod for convection-diffusion problems, Comput Metods Appl Mec Engrg , [3] K. Brix, M. Campos Pinto, W. Damen, A multilevel preconditioner for te interior penalty discontinuous Galerkin metod, SIAM J Numer Anal , [4] P. Castillo, B. Cockburn, I. Perugia, D. Scötzau, An a priori error analysis of te local discontinuous Galerkin metod for elliptic problems, SIAM J Numer Anal 38, [5] B. Cockburn, B. Dong, An analysis of te minimal dissipation local discontinuous Galerkin metod for convection-diffusion problems, J Sci Comput 3 7, [6] B. Cockburn, J. Guzmán, H. Wang, Superconvergent discontinuous Galerkin metods for secondorder elliptic problems, Mat Comput , -4. [7] J. Douglas, T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin metods, Lecture Notes in Pys , 7-6. [8] X. Feng, O.A. Karakasian, Two-level additive Scwarz metods for a discontinuous Galerkin approximation of second order elliptic problems, SIAM J Numer Anal. 394, [9] E.H. Georgoulis, E. Süli, Optimal error estimates for te p-version interior penalty discontinuous Galerkin finite element metod, IMA J Numer Anal 5 5, 5-. [] J. Gopalakrisnan, G. Kanscat, A multilevel discontinuous Galerkin metod, Numer Mat 953 3, [] R.H.W. Hoppe, G. Kanscat, T. Warburton, Convergence analysis of an adaptive interior penalty discontinuous Galerkin metod, SIAM J Numer Anal 47 8, [] G. Kanscat, R. Rannacer, Local error analysis of te interior penalty discontinuous Galerkin metod for second order elliptic problems, J Numer Mat 4, [3] S. Kaya, B. Rivière, A discontinuous subgrid eddy viscosity metod for te time-dependent Navier- Stokes equations, SIAM J Numer Anal , [4] M.G. Larson, A.J. Niklasson, Analysis of a family of discontinuous Galerkin metods for elliptic problems: te one dimensional case, Numer Mat 99 4, 3-3. [5] A. Lasis, E. Süli, p-version discontinuous Galerkin finite element metod for semilinear parabolic problems, SIAM J Numer Anal , [6] J.T. Oden, I. Babuška, C.E. Baumann, A discontinuous p finite element metod for diffusion problems, J Comput Pys , [7] M.R. Om, H.Y. Lee, J.Y. Sin, Error estimates for discontinuous Galerkin metod for nonlinear parabolic equations, J Mat Anal Appl 35 6, [8] M.R. Om, Error estimate for parabolic problem by backward Euler discontinuous Glerkin metod, Int J Appl Mat 9, 7-8. [9] W.H. Reed, T.R. Hill, Triangular mes metods for neutron transport equation, Tecnical Report LA-UR , Los Alamos Scientific Laboratory, 973.

25 IMPLICIT IPDG METHODS FOR NONLINEAR PARABOLIC EQUATIONS 5 [] B. Rivière, M.F. Weeler, V. Girault, Improved energy estimates for interior penalty, constrained and discontinuous Galerkin metods for elliptic problems I, Comput Geosci , [] B. Rivière, M.F. Weeler, V. Girault, A priori error estimates for finite element metods based on discontinuous approximation spaces for elliptic problems, SIAM J Numer Anal 39 3, [] B. Rivière, M.F. Weeler, A discontinuous Galerkin metod applied to nonlinear parabolic equations. Discontinuous Galerkin metods, in: B. Cockburn, G.E. Karaniadakis, C.-W. Su Eds., Discontinuous Galerkin Metods: Teory, Computation and Applications, in: Lecture Notes in Comput. Sci. Engrg., vol., Springer, Berlin,, pp [3] B. Rivière, Discontinuous Galerkin metods for solving elliptic and parabolic equations: teory and implementation, SIAM, Piladelpia, 8. [4] L.J. Song, Fully discrete interior penalty discontinuous Galerkin metods for nonlinear parabolic equations, Numer Met Part D E 8, [5] S. Sun, M.F. Weeler, Symmetric and nonsymmetric discontinuous Galerkin metods for reactive transport in porous media, SIAM J Numer Anal 43 5, [6] R. Temam, Navier-Stokes equations: Teory and Numerical Analysis, Cambridge Press,. [7] K. Wang, H. Wang, S. Sun, M.F. Weeler, An optimal-order L -error estimate for nonsymmetric discontinuous Galerkin metods for a parabolic equation in multiple space dimensions, Comput Metods Appl Mec Engrg 98 9, [8] M.F. Weeler, A priori L error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J Numer Anal 4 973, [9] M.F. Weeler, An elliptic collocation-finite element metod wit interior penalties, SIAM J Numer Anal 5 978, 5-6.