weak Galerkin, finite element methods, interior estimates, second-order elliptic

Transcription

1 INERIOR ENERGY ERROR ESIMAES FOR HE WEAK GALERKIN FINIE ELEMEN MEHOD HENGGUANG LI, LIN MU, AND XIU YE Abstract Consider the Poisson equation in a polytopal domain Ω R d (d = 2, 3) as the model problem We study interior energy error estimates for the weak Galerkin finite element approximation to elliptic boundary value problems In particular, we show that the interior error in the energy norm is bounded by three components: the best local approximation error, the error in negative norms, and the trace error on the element boundaries his implies that the interior convergence rate can be polluted by solution singularities on the domain boundary, even when the solution is smooth in the interior region Numerical results are reported to support the theoretical findings o the best of our knowledge, this is the first local energy error analysis that applies to general meshes consisting of polytopal elements and hanging nodes Key words problems weak Galerkin, finite element methods, interior estimates, second-order elliptic AMS subject classifications Primary: 65N15, 65N30; Secondary: 35J50 1 Introduction Let Ω be a bounded polytopal domain in R d for d = 2, 3 Namely, Ω is a polygon (d = 2) or a polyhedron (d = 3) We consider the Poisson equation with the Dirichlet boundary condition (11) u = f in Ω, u = 0 on Ω, for a given function f L 2 (Ω) he weak Galerkin (WG) finite element method refers to a general finite element technique solving partial differential equations he WG methods are designed to use discontinuous approximations on general polytopal meshes In the presence of discontinuous functions in the finite element approximation, the corresponding discretization formulation is often complicated, and sometimes with undesirable features such as tuning parameters, in order to enforce the interrelation of the finite element solution between elements In contrast, the WG finite element discretization is featured with a simplified formulation that can be derived directly from the weak form of the corresponding equation by replacing strong derivatives with weakly defined derivatives First introduced in [28, 29], the WG method has shown its efficacy in a variety of computational models, such as the Stokes problem, transmission problems, and Maxwell s equations; and its convergence in the global energy norm is well justified (see [21, 22, 20] and references therein) he interior energy error estimate is a critical component for convergence analysis in other norms and is useful in many applications For the usual finite element method, Department of Mathematics, Wayne State University, Detroit, MI 48202, USA (li@wayneedu) his research was supported in part by the National Science Foundation Grant DMS , by the Natural Science Foundation of China Grant , and by the Wayne State University Grants Plus Program Computer Science and Mathematics Division Oak Ridge National Laboratory, Oak Ridge, N, 37831,USA (mul1@ornlgov) his research was supported in part by the US Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under award number ERKJE45; and by the Laboratory Directed Research and Development program at the Oak Ridge National Laboratory, which is operated by U-Battelle, LLC, for the US Department of Energy under Contract DE-AC05-00OR22725 Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA (xxye@ualredu) his research was supported in part by the National Science Foundation Grant DMS

2 2 such interior estimates were established on simplicial elements (see for example [27, 19, 23, 26, 24]) It was shown that the interior energy error of the finite element solution is bounded by the best interior energy approximation error from the finite element space and by the error in low-order norms Different from the best interior approximation error, which depends on the local smoothness of the solution in the interior region, the error in low-order norms is affected by the global smoothness of the solution On polygonal or polyhedral domains, the error in low-order norms can be the dominant term for the interior energy error, due to the solution singularity near corners and edges of the boundary his is referred to as the pollution effect of the finite element method he interior error estimate for the WG method, however, remains an open investigation he goal of this paper is to derive local energy error estimates in an interior region of the domain for the WG approximation to the model problem (11), which can be applied on general polytopal meshes Due to the fact that the WG method utilizes a special variational formulation on discontinuous approximation spaces, the wellestablished local error estimates for the finite element method do not directly apply, and new analysis is needed for further developments o the best of our knowledge, this is also the first interior energy error analysis that applies to general meshes consisting of polytopal elements and hanging nodes In addition, we derive negative norm error estimates for the WG method, which can be of independent interest In equation (11), the solution u H 1 0 (Ω) is unique for f H 1 (Ω) he smoothness of the solution, however, depends on both the given data f and on the geometry of the domain In particular, on polytopal domains, it is well known [17, 13, 8, 16, 6, 11, 9] that the solution generally does not satisfy the full regularity estimate (12) u H m+1 (Ω) C f H m 1 (Ω), m 0, which holds on domains with a smooth boundary his is because the non-smooth boundary points (corners (d = 2) and vertices and edges (d = 3)) can lead to singularities in the solution Consequently, even for a sufficiently smooth function f, the solution can merely belong to H a+1 (Ω), for any 0 < a < η, where η 1/2 is the elliptic regularity index determined by the local geometry of the non-smooth boundary points [10] his lack of regularity can severely slow down the global convergence of the numerical approximation [2, 14, 15, 18, 5, 3, 7, 12, 25] and has been a major concern in the computational community For the WG approximation, it can be shown that the global convergence rate in the energy norm has an upper bound decided by the regularity of the solution (Proposition 22) Meanwhile, as a local property, the regularity of the solution can vary in different local regions of the domain Especially, in an interior region, the solution regularity is decided by the local smoothness of the given function Namely, u H m+1 (G) for f H m 1 (G ) and G G Ω hus, the solution may possess more regularity in the interior region herefore, an intriguing question that is both of theoretical interest and of practical importance is whether in this case the convergence of the WG solution is better in the interior than in the entire domain In this paper, by developing interior energy estimates, we shall establish a connection between the interior error and the global approximation error for the WG method he novelty of the analysis lies in the establishment of a Caccioppoli-type inequality for discrete harmonic functions in the discontinuous WG subspace (see (43)) and in the derivation of the interior negative norm estimate (heorem 37) and of the estimate (415) regarding the bilinear form involving the weak derivatives o be more precise, through a detailed

3 study on the approximation properties of the WG method, we show that the interior energy error is bounded by three components: the best interior energy approximation error, the local negative norms of the error, and the local trace error on the element boundaries (heorem 45) Although the last two errors are of higher order compared with the global energy error, they are not necessarily better than the local energy error from the best interior approximation, especially when the domain is non-convex and the solution possesses singularities Hence, the interior energy approximation of the WG solution can be polluted by the singular solution outside of the region, which resembles the local behavior of the standard finite element method [26] he rest of the paper is organized as follows In Section 2, we define weak functions and introduce the WG finite element scheme hen, we describe global approximation properties of the numerical method for possible singular solutions In Section 3, we obtain the global and interior error analysis for the WG solution in negative norms In Section 4, based on local properties of the WG method and on the negative norm estimates, we shall present our main result on the interior energy error analysis In Section 5, we report numerical test results to validate our theoretical result hroughout the paper, the generic constant C > 0 in our analysis may be different at different occurrences It will depend on the computational domain, but not on the functions involved in the estimates or the mesh size in the numerical algorithms 2 Preliminaries In this section, we introduce necessary notation and define the WG scheme for equation (11) 21 Notation For a bounded domain G R d, we use the conventional notation H m (G), m 0, to denote the Sobolev space consisting of functions whose jth derivatives, 0 j m, are square integrable over G; and L 2 (G) = H 0 (G) he associated semi-norm and norm are v m,g := 1/2 1/2 α v 2 dx, v m,g :=, α =m G j m v 2 j,g where α Z d 0 is the multi-index, namely, α = (α 1, α 2 ) for d = 2 and α = (α 1, α 2, α 3 ) for d = 3, and α := 1 i d α i In addition, for s R +, we define the space H s (G) by interpolation Let H s 0(G) be the closure of C 0 (G) in H s (G) hen, the space H s (G) is defined to be the dual space of H s 0(G) (Chapter III in [1]) hen, we start with the definition of weak functions Definition 21 (Weak Functions) On a polytopal domain K, a weak function refers to the collection of two functions v = {v 0, v b } such that v 0 L 2 (K) and v b H 1/2 (e), where e is any edge or flat face of K herefore, v 0 is understood as the value of v in K and v b represents v on K Note that even if the trace of v 0 is well defined on the boundary, v b may not necessarily be related to v 0 on K We denote the space of weak functions on K by W (K) = {v = {v 0, v b } : v 0 H 1 (K), v b H 1/2 (e) for each edge or flat face e K} he Sobolev space H 1 (K) can be embedded into the space W (K) by the inclusion map 3 i W (v) = {v K, v K } v H 1 (K) herefore, the space H 1 (K) can be regarded as a subspace of W (K) by identifying each v H 1 (K) with i W (v) On the other hand, a weak function {v 0, v b } W (K)

4 4 is said to be in H 1 (K) if it can be identified with a function v H 1 (K) using the inclusion map Using the notation introduced above, we can formulate a discrete space for the WG finite element approximation Let h be a partition of the domain Ω consisting of shape-regular polygons (Ω R 2 ) or polyhedra (Ω R 3 ) satisfying a set of conditions specified in [29] Denote by E h the set of all edges or flat faces in h For every element h, we denote by h its diameter and the mesh size is h = max h h for h o simplify the presentation, we assume there exists C > 0 independent of h, such that h C min h h For a given integer k 1 and a region G Ω, denote by P k (G) the space of polynomials of degree k on G hen, we define the space of weak functions associated with h, V = {v = {v 0, v b } : v 0 H 1 ( ), v b e H 1/2 (e), e E h, h ; v b = 0 on Ω}, and define the WG finite element space associated with h, V h = {v = {v 0, v b } : v 0 P k ( ), v b e P k (e), e, h ; v b = 0 on Ω} We emphasize that any v V h has a single value v b on each e E h In addition, for any G Ω, let V G,h be the restriction of V h on G, and define (21) V G,h = {v V G,h : supp v G} It is clear that V G,h V h hen, we define the weak gradient for weak functions Definition 22 For h and a weak function v = {v 0, v b } W ( ), a weak gradient w v [P k 1 ( )] d on is defined by (22) ( w v, τ) = (v 0, τ) + v b, τ n, τ [P k 1 ( )] d, where (v, w) = vwdx, v, w = vwds, and n is the outward normal direction to Next, we will introduce three global projections Q 0, Q b and Q h hey are elementwise defined L 2 projections detailed as follows For each element h, Q 0 : L 2 ( ) P k ( ) and Q b : L 2 (e) P k (e) are the L 2 projections onto the associated local polynomial spaces, Q h is the L 2 projection from [L 2 ( )] d to the local weak gradient space [P k 1 ( )] d hen, we define the projection operator Q h : V V h, such that on each element h Q h v = {Q 0 v 0, Q b v b }, for any v = {v 0, v b } V If a function v = {v 0, v b } is smoother such that for any e E h, v 0 e = v b e, we simplify the notation for Q h v by Q h v = {Q 0 v, Q b v} Note that Q h and Q h are locally defined operators We further assume the partition h gives rise to the optimal local convergence rates (23) (25) and the standard inverse inequality (26) Namely, for 1/2 < a k, the following approximation properties hold (23) (24) v Q 0 v 0, + h (v Q 0 v) 0, Ch a+1 v a+1,, v H a+1 ( ), v Q b v 0,e Ch a+1 v a+1,e, v H a+1 (e),

5 5 and for 0 a 1 1 and v [H a2 ( )] d, where a 1 < a 2 k, we have (25) v Q h v a1, Ch a2 a1 v a2, In addition, for v V h, (26) v l, Ch l v 0,, 0 l k Examples of such partitions include shape-regular triangular or tetrahedral triangulations, and polytopal partitions in [22, 29] Note that the estimate (25) is a consequence of the operator-interpolation theory (Chapter 14 in [4]) and the approximation property (Lemma 41 in [29]) v Q h v 0, + h v Q h v 1, Ch k v k, 22 Weak Galerkin finite element schemes We introduce necessary bilinear forms in order to formulate the numerical scheme For v, w V, define s (v, w) = h 1 v 0 v b, w 0 w b, a (v, w) = ( w v, w w) + s (v, w); and (27) s(v, w) = s (v, w), a(v, w) = a (v, w) h h For G Ω, we denote by s G (, ) the restriction of s(, ) on G hen, we define the WG approximation of equation (11) as in [21] Weak Galerkin Algorithm 1 A unique numerical approximation for (11) can be obtained by seeking u h = {u 0, u b } V h that satisfies the following equation: (28) a(u h, v) = (f, v 0 ), v = {v 0, v b } V h For v V, we define a norm associated to the bilinear form (29) v Ω = a(v, v) In addition, for v V h + H 1 0 (Ω), define the discrete H 1 energy norm (210) v h,ω = ( h ( v 0 2 0, + h 1 v 0 v b 2 0, ) ) 1/2 In particular, if v H 1 (Ω), it can be seen that v h,ω = v 0,Ω For a subregion G Ω, we denote by h,g and G the restrictions of h,ω and Ω on G, respectively Moreover, the following discrete Poincaré inequality holds for functions in V h (Lemma 34 in [20]) Proposition 21 For v = {v 0, v b } V h, we have h v 0 2 0, + h v 0 v b 2 C w v 2 0,Ω

6 6 Before proceeding with our interior analysis, we recall the global error estimates regarding the WG finite element method Proposition 22 Let e h = Q h u u h, where u h = {u 0, u b } is the WG finite element solution arising from (28) hen, for any v V h we have (211) a(e h, v) = l u (v) + s(q h u, v), where l u (v) = h ( u Q h u) n, v 0 v b Consequently, if the exact solution u H µ+1 (Ω) for µ > 0, we have (212) u u h h,ω u Q h u h,ω + u h Q h u Ω Ch t u t+1,ω, where t = min(µ, k) In addition, for any v V h, let G be a region such that supp v G Ω Using the same notation µ and t, we have (213) (214) s(q h u, v) Ch t s G (v, v) 1/2 u t+1,g Ch t u t+1,g v G, l u (v) Ch t u t+1,g v G Proof he error equation (211) can be found in [21] he estimates (213) and (214) follow from the approximation properties (23) (25) and from the same lines as in Lemma 54, [22] he energy norm estimate (212) can be derived using the error equation (211) and the estimates in (213) and (214) Remark 23 he global estimates in Proposition 22 were derived based on the local approximation properties associated to the L 2 projection operators Q h and Q h for the solution u H µ+1 (Ω) It is well known that the solution regularity in equation (11) depends on the given function f and on the geometry of the domain For example, on a polytopal domain, the solution can possess singularities near nonsmooth points on the boundary, and the full regularity estimate (12) does not hold for all m 0 In fact, there is a regularity index η 1/2 that is determined by the domain geometry, such that u only belongs to H a+1 (Ω) for 0 < a < η, even when the given data is smooth (see Chapter 2 in [10]), and it holds that u a+1,ω C f a 1,Ω In particular, on a polygon, η = π/χ, where χ is the largest interior angle of the domain Meanwhile, the regularity is a local property In an interior region G Ω, the solution regularity is determined by the given function f (see [10]), since it is away from the boundary herefore, the solution of equation (11) may possess different smoothness in G than other parts of the domain Our study shall shed light on local energy error analysis of the WG method that applies to general meshes consisting of polytopal elements and hanging nodes 3 Negative norm estimates In this section, we derive global and interior negative norm estimates for the WG method 31 Some lemmas Let h be an element and let e E h be such that e We first recall the trace inequality [29] for a function ϕ H 1 ( ), (31) ϕ 2 e C ( h 1 ϕ 2 + h ϕ 2 )

7 For functions in V h, the gradient operator and the weak gradient operator are connected through (22), and satisfy the following estimates Lemma 31 For v V h and h, we have ( ) (32) v 0 C w v + h 1/2 v 0 v b, ( ) (33) w v C v 0 + h 1/2 v 0 v b 7 Proof Let τ = v 0 in (22) Using integration by parts, we obtain ( w v, v 0 ) = ( v 0, v 0 ) + v b v 0, v 0 n Using the trace estimate (31) and the inverse inequality, we have v 0 2 w v v 0 + v 0 v b v 0 w v v 0 + h 1/2 v 0 v b h 1/2 (h 1/2 v 0 + h 1/2 v 0 1, ) = w v v 0 + Ch 1/2 v 0 v b v 0, which leads to ( ) v 0 C w v + h 1/2 v 0 v b he estimate (33) follows from a similar calculation In addition, the L 2 projection Q h possesses the following local properties Lemma 32 Let h and v H 1 ( ) hen, we have (34) v Q b v 0, v Q 0 v 0, Let ω C ( ) such that D l ω < C 0 for 0 l k + 1, where C 0 is a constant independent of h hen, for e E h with e, we have (35) (36) h 1 ωv Q 0(ωv) 0, + (ωv Q 0 (ωv)) 0, Ch v 1,, v P k ( ), ωv Q b (ωv) 0,e Ch v 0,e, v P k (e) Proof he estimate (34) follows from the following calculation v Q b v 2 0, = v Q b v, v Q b v = v Q b v, v Q 0 v v Q b v 0, v Q 0 v 0, For v P k ( ) and α = k +1, note that α v = 0 on Recall the approximation property in (23) hen, using the inverse inequality, we have h 2 ωv Q 0(ωv) 2 0, + (ωv Q 0 (ωv)) 2 0, Ch 2k ωv 2 k+1, Ch 2k ωv 2 l, Ch 2k v 2 l, Ch 2 v 2 1,, 0 l k+1 0 l k which proves the estimate (35) he estimate (36) follows from a similar calculation based on the approximation property (24) he estimates (35) and (36) do not include high-order norms on the right hand side and represent certain super-approximation properties for the functions involved

8 8 In addition, we have a useful property regarding the L 2 projection operators Q h and Q h in Definition 22 Lemma 33 For each h and v H 1 ( ), we have w (Q h v) = Q h ( v) Proof Following the definition in (22), we have for any τ [P k 1 ( )] d, ( w (Q h v), τ) = (Q 0 v, τ) + Q b v, τ n = (v, τ) + v, τ n = ( v, τ) = (Q h ( v), τ), which completes the proof 32 Global negative norm analysis Consider the auxiliary problem (37) φ = g in Ω, φ = 0 on Ω, where g C 0 (Ω) herefore, according to Remark 23, (38) φ a+1,ω C g a 1,Ω, 0 a < η, where η > 1/2 is the regularity index hen, using a duality argument, we obtain the global negative norm estimate for the WG solution heorem 34 Suppose the solution of equation (11) u H µ+1 (Ω), for some µ > 1/2 Let t = min(µ, k) and p = min(a, k), where a is the index in (38) hen, for l 0, the WG solution u h = {u 0, u b } satisfies { Ch (39) u u 0 l,ω t+l+1 u t+1,ω if 0 l p 1, Ch t+p u t+1,ω if l > p 1 Proof Let e 0 = Q 0 u u 0, e b = Q b u u b, and e h = {e 0, e b } Note that (310) u u 0 l,ω u Q 0 u l,ω + Q 0 u u 0 l,ω Since Q 0 is the local L 2 projection on each element, we have (311) u Q 0 u l,ω = (u Q 0 u, g) (u Q 0 u, g Q 0 g) sup = sup g 0 C0 (Ω) g l,ω g 0 C0 (Ω) g l,ω Ch t+l+1 u t+1,ω if 0 l k + 1, or Ch t+k+2 u t+1,ω if l > k + 1 For the second term in (310), using (37), we obtain (312) e 0 l,ω = (e 0, g) sup = g 0 C0 (Ω) g l,ω sup g 0 C 0 (Ω) (e 0, φ) g l,ω Using integration by parts and e b Ω = 0, we have (e 0, φ) = (313) ( φ, e 0 ) φ n, e 0 e b h h

9 9 With the estimate in Lemma 33, (22), and integration by parts, it holds that ( w Q h φ, w e h ) = (Q h ( φ), w e h ) = (e 0, (Q h φ)) + e b, Q h ( φ) n (314) = ( e 0, Q h φ) e 0 e b, Q h ( φ) n = ( φ, e 0 ) e 0 e b, Q h ( φ) n hus, by (313), (314), and (211), we have (e 0, φ) = ( ) ( w Q h φ, w e h ) + (Q h ( φ) φ) n, e 0 e b (315) h = l u (Q h φ) + s(q h u, Q h φ) s(e h, Q h φ) l φ (e h ), where l u (Q h φ) is defined as in Proposition 22 hen, by the triangle inequality, (316) l u (Q h φ) ( u Q h u) n, Q 0 φ φ h + ( u Q h u) n, Q b φ φ h Note that Q b φ Ω = φ Ω = 0 herefore, h ( u Q h u) n, Q b φ φ = Meanwhile, using the trace estimate (31), we have h u n, Q b φ φ = 0 Q 0 φ φ 0, C(h 1/2 Q 0 φ φ 0, + h 1/2 (Q 0φ φ) 0, ) herefore, using the trace estimate, (23), and (25), we have (317) (318) (319) Q 0 φ φ 0, Ch 3/2+l φ l+2, for 0 l p 1, Q 0 φ φ 0, Ch 1/2+p φ p+1, for l > p 1, u Q h u 0, Ch t 1/2 u t+1, We briefly show the estimate (319) Suppose a vertex of h lies at the origin Let ˆ = h 1 be the reference polytopal obtained after the scaling with factor h 1 at the origin According to the trace estimate [10], for v H 1/2+ɛ ( ˆ ) with ɛ > 0 arbitrarily small, v L 2 (ê) C( v L 2 ( ˆ ) + v H 1/2+ɛ ( ˆ ) ), where v H 1/2+ɛ ( ˆ ) ( = v(x) v(y) 2 dxdy ) 1/2 ˆ ˆ x y is the H 1/2+ɛ ( ˆ ) semi-norm, and ê = h 1 d+1+2ɛ e for any e E h and e Using the scaling argument, we have v L2 (e) C(h 1/2 v L 2 ( ) + h ɛ v H 1/2+ɛ ( )) C(h 1/2 v L 2 ( ) + h ɛ v H 1/2+ɛ ( )) Recall t > 1/2 Replacing v by u Q h u and using the estimate in (25), we obtain the inequality in (319) herefore, by (316) (319) and the Cauchy-Schwarz inequality, it follows that (320) l u (Q h φ) C ( u Q h u 2 ) 1/2 ( 0, Q 0 φ φ 2 0, h h C(L )h t u t+1,ω, ) 1/2

10 10 where (321) L := { h l+1 φ l+2,ω if 0 l p 1, h p φ p+1,ω if l > p 1 Similarly, by the definition of Q b, the trace estimate (31), and (23), we have (322) s(q h u, Q h φ) h 1 h h 1 h C ( h 1 h C(L )h t u t+1,ω Q b(q 0 u u) 0, Q b (Q 0 φ φ) 0, Q 0u u 0, Q 0 φ φ 0, Q 0u u 2 ) 1/2 ( 0, h 1 h Q 0φ φ 2 ) 1/2 0, For the third term in (315), using a similar calculation as above, by (33), the estimate in Proposition 22, (23), and (24), we have (323) s(e h, Q h φ) C ( h h 1 Q 0φ φ 2 0, ) 1/2 eh Ω C(L ) e h h,ω C(L )( u u h h,ω + u Q h u h,ω ) C(L )h t u t+1,ω For the last term in (315), by the trace estimate, Proposition 22, and (319), we have (324) l φ (e h ) = (Q h ( φ) φ) n, e 0 e b h C ( Q h ( φ) φ 2 ) 1/2h 0, 1/2 e h Ω C(L )h t u t+1,ω h hen, by (315) and (320) (324), we obtain (325) (e 0, φ) herefore, according to (38), we obtain { h t+l+1 u t+1,ω φ l+2,ω if 0 l p 1, h t+p u t+1,ω φ p+1,ω if l > p 1 (326) (327) φ l+2,ω C g l,ω if 0 l p 1, φ p+1,ω C g p 1,Ω C g l,ω if l > p 1 he proof is completed by combining (310) (312) and (325) (327) 33 Interior negative norm analysis We start to analyze the negative norm of the error in an interior region o fix the notation, we carry out the analysis on d- dimensional concentric balls D i, 0 i 3, such that D 0 D 1 D 2 D 3 Ω with radii r i In the text below, we assume the mesh parameter h is sufficiently small, such that for 0 i 2, dist( D i, D i+1 ), dist(d 3, Ω) k 0 h, where k 0 is a positive constant For G Ω, let G = { i } be the set of elements such that i h and i G Denote by G = G herefore, for any v V G,h (see (21)), v Ω\ G = 0 o study

11 the local error, we introduce the following cutoff smooth functions For 1 i 3, let ω i C 0 (D i ) be such that (328) ω i = 1 in D i hen, when h is sufficiently small, we have D i D i+1 D i+1, 0 i 2 Let S(ω 2 u) = {[S(ω 2 u)] 0, [S(ω 2 u)] b } V D3,h be the WG approximation of ω 2 u on D 3, namely (329) a(s(ω 2 u), v) = ( (ω 2 u), v 0 ), v V D3,h In what follows, we redefine the auxiliary equation (37), such that Ω = D 2 and g C 0 (D 2 ) Since the ball D 2 has a smooth boundary, it holds for any l 0 that 11 (330) φ l+2,d2 C g l,d2 Recall the WG solution of equation (11) u h = {u 0, u b } hen, we derive the following equation Lemma 35 Let v 0 = u u 0 hen, for h sufficiently small, we have (ω 1 v 0, φ) D2 = (v 0, (φ ω 1 )) D2 + ( φ, v 0 ω 1 ) D2 + (v 0, (ω 1 φ)) D2 Proof Using supp(ω 1 ) D 2 and integration by parts, we have (ω 1 v 0, φ) D2 = ω 1 v 0, φ n + ( φ, ω 1 v 0 + v 0 ω 1 ) = ( ) ( (ω1 φ), v 0 ) (φ ω 1, v 0 ) + ( φ, v 0 ω 1 ) ω 1 v 0, φ n = ( ( (ω1 φ), v 0 ) + (v 0, (φ ω 1 )) + ( φ, v 0 ω 1 ) v 0, (ω 1 φ) n ) = (v 0, (φ ω 1 )) D2 + ( φ, v 0 ω 1 ) D2 + (v 0, (ω 1 φ)) D2, which completes the proof Next, we derive an estimate regarding Q 0 (ω 2 u) [S(ω 2 u)] 0 Lemma 36 Define q h = {q 0, q b } = Q h (ω 2 u) S(ω 2 u) V h Suppose u H µ+1 (D 2 ) for some µ > 1/2 Let t = min(µ, k) For l 0, let l k = min(l, k 1) hen, for h sufficiently small, we have (q 0, (ω 1 φ)) D2 Ch t+l k+1 u t+1,d2 φ l+2,d1 Proof Using integration by parts and noting ω 1 = 0, we have D2 (331) (q 0, (ω 1 φ)) D2 = ( (ω 1 φ), q 0 ) (ω 1 φ) n, q 0 q b Replacing φ by ω 1 φ and e h by q h in (314), we have (332) ( w Q h (ω 1 φ), w q h ) = ( (ω 1 φ), q 0 ) q 0 q b, Q h ( (ω 1 φ)) n

12 12 Using (332), (331), (329), and (211), we further obtain (q 0, (ω 1 φ)) D2 = ( w Q h (ω 1 φ), w q h ) D2 + (333) (Q h ( (ω 1 φ)) (ω 1 φ)) n, q 0 q b = ( w Q h (ω 1 φ), w q h ) D3 + = l ω2u(q h (ω 1 φ)) D3 + s D3 (Q h (ω 2 u), Q h (ω 1 φ)) D3 (Q h ( (ω 1 φ)) (ω 1 φ)) n, q 0 q b s D3 (q h, Q h (ω 1 φ)) l ω1φ(q h ) D3 Recall φ H l+2 (D 2 ) for any l 0 (see (330)) Replacing u by ω 2 u and φ by ω 1 φ in (316), (320), and noting that Q b (ω 1 φ) D3 = 0, we have (334) l ω2u(q h (ω 1 φ)) D3 Ch t+l k+1 ω 2 u t+1, D3 ω 1 φ lk +2, D 3 Ch t+l k+1 u t+1,d2 φ l+2,d1 Similarly, replacing u by ω 2 u and φ by ω 1 φ in (322), we have (335) s D3 (Q h (ω 2 u), Q h (ω 1 φ)) Ch t+l k+1 ω 2 u t+1, D3 ω 1 φ lk +2, D 3 Ch t+l k+1 u t+1,d2 φ l+2,d1 Replacing u by ω 2 u, φ by ω 1 φ, and e h by q h in (323), it follows that (336) s D3 (q h, Q h (ω 1 φ)) Ch t+l k+1 u t+1,d2 φ l+2,d1 Replacing u by ω 2 u, φ by ω 1 φ, and e h by q h in (324), we obtain (337) l ω1φ(q h ) D3 Ch t+l k+1 u t+1,d2 φ l+2,d1 Hence, the proof is completed by combining the estimates (333) (337) hen, we present our interior negative norm estimate for the WG approximation heorem 37 Recall the concentric balls D 0 D 1 D 2 D 3 Ω Suppose the solution of equation (11) u H µ+1 (D 2 ) for some µ > 1/2 Let t = min(µ, k) For l 0, let l k = min(l, k 1) hen, for h sufficiently small, we have u u 0 l,d0 C(h t+l k+1 u t+1,d2 + h l k+1 s D2 (u h, u h ) 1/2 + u u 0 l 1,D2 ), where u h = {u 0, u b } is the WG solution defined in (28) Proof Recall the redefined auxiliary equation φ = g C0 (D 2 ) with the Dirichlet boundary condition and its regularity estimate (330) hen, by the definitions of negative norms and the cutoff function ω 1, we have u u 0 l,d0 = ω 1 (u u 0 ) l,d0 ω 1 (u u 0 ) l,d2 (338) (ω 1 (u u 0 ), g) D2 = sup g 0 C0 (D2) g l,d2 = sup g 0 C 0 (ω 1 (u u 0 ), φ) D2 (D2) g l,d2 Let v 0 = u u 0 By Lemma 35 and the boundedness of ω 1, we have (ω 1 v 0, φ) D2 = (v 0, (φ ω 1 )) D2 + ( φ, v 0 ω 1 ) D2 + (v 0, (ω 1 φ)) D2 (339) C v 0 l 1, φ D2 l+2, + (v D2 0, (ω 1 φ)) D2

13 13 We write (340) v 0 = u Q 0 (ω 2 u) + Q 0 (ω 2 u) [S(ω 2 u)] 0 + [S(ω 2 u)] 0 u 0 In view of (339), we shall analyze these terms separately By the definition of Q 0, supp(ω 1 ) D 1, and the boundedness of ω 2, we have (u Q 0 (ω 2 u), (ω 1 φ)) D2 = (ω 2 u Q 0 (ω 2 u), (ω 1 φ)) D1 (341) = (ω 2 u Q 0 (ω 2 u), (ω 1 φ) + Q 0 ( (ω 1 φ))) D2 Ch t+l k+1 u t+1, D2 φ l+2,d1 Let w h = S(ω 2 u) u h = {w 0, w b } Note that ω 1 = 0 herefore, using D2 integration by parts and equation (314), we have (w 0, (ω 1 φ)) D2 = ( ) ( (ω1 φ), w 0 ) w 0, (ω 1 φ) n = = ( ( (ω1 φ), w 0 ) w 0 w b, (ω 1 φ) n ) ( ( (ω1 φ), w 0 ) w 0 w b, Q h ( (ω 1 φ)) n (342) = ) w 0 w b, [ (ω 1 φ) Q h ( (ω 1 φ))] n ( ) ( w Q h (ω 1 φ), w w h ) w 0 w b, [ (ω 1 φ) Q h ( (ω 1 φ))] n By the trace estimate (31), (319), and the boundedness of ω 1, it follows that w 0 w b, [ (ω 1 φ) Q h ( (ω 1 φ))] n C w 0 w b 0, h l k+1 ω 1 φ lk +2, (343) h 1/2 Ch l k+1 s D2 (w h, w h ) 1/2 φ lk +2,D 1 Meanwhile, since Q h (ω 1 φ) V D2,h V D3,h, by (329), (28), and ω 2 D2 = 1, we have a(q h (ω 1 φ), w h ) = a(q h (ω 1 φ), S(ω 2 u) u h ) h 1 = ( u + (ω 2 u), Q 0 (ω 1 φ)) = 0 herefore, by the estimate above, Q b Q 0 e = Q 0 e for e E h, the trace estimate, and (23), we have ( w Q h (ω 1 φ), w w h ) = h 1 w 0 w b, Q 0 (ω 1 φ) Q b (ω 1 φ) = h 1 w 0 w b, Q 0 (ω 1 φ) ω 1 φ C ( w 0 w b 2 ) 1/2 ( 0, Q 0(ω 1 φ) ω 1 φ 2 ) 1/2 0, (344) Ch l k+1 s D2 (w h, w h ) 1/2 φ lk +2,D 1 h 1

14 14 By ω 2 D2 = 1 and Proposition 22, it holds that s D2 (w h, w h ) 1/2 ω 2 u S(ω 2 u) h, + s D3 D2 (u u h, u u h ) 1/2 (345) Ch t u t+1,d2 + s D2 (u h, u h ) 1/2 Hence, by (338) (345), Lemma 36, and the regularity estimate (330), we obtain u u 0 l,d0 C(h t+l k+1 u t+1,d2 + h l k+1 s D2 (u h, u h ) 1/2 + u u 0 l 1,D2 ), which completes the proof 4 Interior Energy Estimates In this section, we derive our main results for the WG finite element solution in interior regions of the computational domain In particular, for D 0 Ω, we obtain estimates on the local error in the energy norm u u h h,d0 We fix the notation by carrying out the analysis on the concentric balls D 0 D 1 D 2 D 3 D Ω he estimate on general interior regions will follow from a covering argument Note that based on the definition in (328), we have (41) u u h h,d0 = ω 1 u u h h,d0 ω 1 u Q h (ω 1 u) h, D2 + Q h (ω 1 u) u h h,d0 Due to (23) and (24), the first term in (41) represents the best approximation error to ω 1 u in an interior region D 2 Hence, we shall focus on the analysis for the second term Q h (ω 1 u) u h h,d0 Note that for v = {v 0, v b }, w = {w 0, w b } V h and h, we have (v, w) = (v 0, w 0 ) and v = v 0 hen, we have the following lemma Lemma 41 Recall the local discrete space in (21) Using the bilinear form a(, ) in (27), we define the projection P D2 onto V D2,h, such that for v V h, (42) a(ω 1 v P D2 (ω 1 v), w) = 0, w V D2,h hen, for h sufficiently small, we have (43) ω 1 v P D2 (ω 1 v) h, D2 + ω 1 v Q h (ω 1 v) D2 Ch v D2 Proof Based on the definition (29), supp ω 1 D 2, and (42), for any w V D2,h, we have ω 1 v P D2 (ω 1 v) 2 D2 = a(ω 1 v P D2 (ω 1 v), ω 1 v P D2 (ω 1 v)) = a(ω 1 v P D2 (ω 1 v), ω 1 v w) herefore, taking w = Q h (ω 1 v) V D2,h, we have ω 1 v P D2 (ω 1 v) D2 ω 1 v w D2 (44) ω 1 v P D2 (ω 1 v) D2 ω 1 v Q h (ω 1 v) D2 By the definition in (328), there exists C > 0 independent of h, such that D l ω 1 < C for 0 l k + 1 herefore, for D2, by the definitions in (210) and (29),

15 15 and by (35), we have ω 1 v P D2 (ω 1 v) h, (ω 1 v P D2 (ω 1 v)) 0, + ω 1 v P D2 (ω 1 v) (ω 1 v Q h (ω 1 v)) 0, + (Q h (ω 1 v) P D2 (ω 1 v)) 0, + ω 1 v P D2 (ω 1 v) Ch v 0 1, + (Q h (ω 1 v) P D2 (ω 1 v)) 0, + ω 1 v P D2 (ω 1 v) aking the summation of the above inequality over all D2, by supp ω 1 D 1, (32), and (44), we have ω 1 v P D2 (ω 1 v) h, D2 C ( h( (45) v 0 2 1, ) 1/2 + Q h (ω 1 v) P D2 (ω 1 v) D2 + ω 1 v P D2 (ω 1 v) D2 C ( h( ) v 0 2 1, ) 1/2 + Q h (ω 1 v) ω 1 v D2 + ω 1 v P D2 (ω 1 v) D2 C ( h( v 0 2 1, ) 1/2 ) + Q h (ω 1 v) ω 1 v D2 Let ψ = Q h (ω 1 v) ω 1 v Recall w ψ [P k 1 ( )] d and (46) ψ 2 D2 = ( w ψ, w ψ) + h 1 ψ 0 ψ b, ψ 0 ψ b hen, by (22), the trace estimate (31) and the inverse inequality, we have w ψ 0, = (47) ( w ψ, q) ( ψ 0, q) + ψ b ψ 0, q n sup = sup 0 q [P k 1 ( )] q d 0, 0 q [P k 1 ( )] q d 0, ψ 0 0, + C ψ b ψ 0 0, (h 1/2 q 0, + h 1/2 q 0, ) q 0, ψ 0 0, + Ch 1/2 ψ b ψ 0 0, hen, by (34), (36), the boundedness of ω 1, the trace estimate, (35), and the inverse inequality, we further have ψ b ψ 0 0, = Q b (ω 1 v b ) ω 1 v b (Q 0 (ω 1 v 0 ) ω 1 v 0 ) 0, ω 1 v 0 ω 1 v b Q b (ω 1 v 0 ω 1 v b ) 0, + ω 1 v 0 Q b (ω 1 v 0 ) 0, + ω 1 v 0 Q 0 (ω 1 v 0 ) 0, C(h v 0 v b 0, + ω 1 v 0 Q 0 (ω 1 v 0 ) 0, ) C(h v 0 v b 0, + h 1/2 ω 1 v 0 Q 0 (ω 1 v 0 ) 0, +h 1/2 ω 1v 0 Q 0 (ω 1 v 0 ) 0, ) C(h v 0 v b 0, + h 3/2 v 0 1, ) hus, by (45) (47), (35), (32), and Proposition 21, we have (ω 1 v P D2 (ω 1 v)) h, D2 + ω 1 v Q h (ω 1 v) D2 C ( h 2 v 0 2 1, + ψ 0 2 0, + h v 0 v b 2 0, + h 2 v 0 2 1, Ch ( v 0 2 1, + h 1 v 0 v b 2 ) 1/2 0, Ch v D2, ) 1/2 )

16 16 which completes the proof Next, we derive an equation regarding Q h (ω 3 u) u h that will be useful to carry out further analysis Lemma 42 Define z = Q h (ω 3 u) u h V h For h and v V h, we have ( w (ω 1 z), w v) = ( w z, w (ω 1 v)) + (z 0 ω 1, v 0 ) + (z 0, (v 0 ω 1 )) (48) z 0, v 0 ω 1 n + ζ (z, v, ω 1 ) + l (z, v, ω 1 ), where ζ (z, v, ω 1 ) = v b v 0, Q h (ω 1 z 0 ) n ω 1 w z n, l (z, v, ω 1 ) = z b z 0, ω 1 w v n Q h (ω 1 v 0 ) n (49) Proof It follows from the product rule and integration by parts that ( (ω 1 z 0 ), v 0 ) = ( z 0, (ω 1 v 0 )) + (z 0 ω 1, v 0 ) + (z 0, (v 0 ω 1 )) z 0, v 0 ω 1 n Using (22) and integration by parts, we have for w W ( ) and v V h, ( w w, w v) = (w 0, w v) + w b, w v n = ( w 0, w v) + w b w 0, w v n = (v 0, Q h w 0 ) + v b, Q h w 0 n + w b w 0, w v n (410) = ( w 0, v 0 ) + v b v 0, Q h w 0 n + w b w 0, w v n Letting w = ω 1 z in (410), we get (411) ( w (ω 1 z), w v) = ( (ω 1 z 0 ), v 0 ) + v b v 0, Q h (ω 1 z 0 ) n + z b z 0, ω 1 w v n Similarly, we have (412) ( w z, w (ω 1 v)) = ( z 0, (ω 1 v 0 )) + z b z 0, Q h (ω 1 v 0 ) n + v b v 0, ω 1 w z n herefore, by (411), (49), and (412), we obtain ( w (ω 1 z), w v) = ( (ω 1 z 0 ), v 0 ) + v b v 0, Q h (ω 1 z 0 ) n + z b z 0, ω 1 w v n = ( z 0, (ω 1 v 0 )) + (z 0 ω 1, v 0 ) + (z 0, (v 0 ω 1 )) z 0, v 0 ω 1 n + v b v 0, Q h (ω 1 z 0 ) n + z b z 0, ω 1 w v n = ( w z, w (ω 1 v)) z b z 0, Q h (ω 1 v 0 ) n v b v 0, ω 1 w z n + (z 0 ω 1, v 0 ) + (z 0, (v 0 ω 1 )) z 0, v 0 ω 1 n + v b v 0, Q h (ω 1 z 0 ) n + z b z 0, ω 1 w v n = ( w z, w (ω 1 v)) + (z 0 ω 1, v 0 ) + (z 0, (v 0 ω 1 )) z 0, v 0 ω 1 n + ζ (z, v, ω 1 ) + l (z, v, ω 1 )

17 his completes the proof In addition, we obtain an estimate on the difference between the weak gradient and the usual gradient Lemma 43 For h and v V h, we have (413) w v v 0 Ch 1/2 v 0 v b Proof It follows from the definition of weak gradient (22) that for any q [P k 1 ( )] d, (414) ( w v, q) = (v 0, q) + v b, q n = ( v 0, q) + v b v 0, q n Letting q = w v v 0 in the above equation and using the trace inequality (31) and the inverse inequality, we have which proves the lemma w v v 0 2 = v b v 0, ( w v v 0 ) n Ch 1/2 v 0 v b w v v 0, With the estimates developed above, we proceed to analyze the local error u u h h,d0 in the interior of the domain Recall z = Q h (ω 3 u) u h in Lemma 42 hen, we further have the following estimate on z Lemma 44 Recall the bilinear form s(, ) in (27) and its restriction s D2 (, ) on D 2 Suppose u H µ+1 (D 2 ) for some µ > 1/2 Let t = min(µ, k) and h be sufficiently small hen, for any v V D2,h, we have (415) a(ω 1 z, v) C(h t u t+1,d2 + u u 0 0,D2 + hs D2 (u h, u h ) 1/2 ) v D2, where u h = {u 0, u b } V h is the WG approximation in (28) he proof of the lemma is long and can be found in Appendix Next, we are ready to present the interior energy estimate for the WG finite element approximation heorem 45 Let u and u h = {u 0, u b } V h be the solution of equation (11) and its WG approximation (28), respectively Let D 0 D Ω be interior regions Suppose the solution of equation (11) u H µ+1 (D) for some µ > 1/2 Let t = min(µ, k) hen, for 0 l k 1 and for h sufficiently small, we have u u h h,d0 C(h t u t+1,d + 17 l h l j u u 0 j,d + h l+1 s D (u u h, u u h ) 1/2 ) j=0 Proof We show the proof for concentric balls D 0 D 1 D 2 D 3 D he case for a general interior region follows from a covering argument using these balls By the definition (328), we have Q h (ω 1 u) D0 = Q h (ω 3 u) D0 Following (41) and using (212) for ω 1 u Q h (ω 1 u) h,, by the boundedness of ω D2 1, we obtain (416) u u h h,d0 ω 1 u Q h (ω 1 u) h, + Q D2 h (ω 3 u) u h h,d0 Ch t u t+1,d2 + Q h (ω 3 u) u h h,d0

18 18 Recall z = Q h (ω 3 u) u h in Lemma 44 It follows from (43), (32), (33), the inverse inequality, (A11), and (A12) that z h,d0 = ω 1 z h,d0 ω 1 z P D2 (ω 1 z) h,d0 + P D2 (ω 1 z) h,d0 Ch z D2 + P D2 (ω 1 z) h,d0 (417) C z 0, D2 + hs D2 (z, z) 1/2 + P D2 (ω 1 z) D2 C(h t+1 u t+1,d2 + u u 0 0,D2 + hs D2 (u h, u h ) 1/2 ) + P D2 (ω 1 z) D2 Using (415), we have (418) a(p D2 (ω 1 z), v) a(ω 1 z, v) P D2 (ω 1 z) D2 = sup = sup 0 v V D2 v,h D2 0 v V D2 v,h D2 C(h t u t+1,d2 + u u 0 0,D2 + hs D2 (u h, u h ) 1/2 ) hus, by (416) (418), we derive (419) u u h h,d0 C(h t u t+1,d2 + u u 0 0,D2 + hs D2 (u h, u h ) 1/2 ) For any l 0, we introduce a set of concentric balls D 2 D 2 D 2 D 2 D 3 D l+2 = D Note that s D2 (u h, u h ) 1/2 = s D2 (u u h, u u h ) 1/2 is part of the local energy error in the interior region D 2 hen, the estimate (419) leads to (420) hs D2 (u u h, u u h ) 1/2 Ch(h t u t+1,d 2 + u u 0 0,D 2 + hs D 2 (u h, u h ) 1/2 ) By (420) and the interior negative norm estimate in heorem 37, it holds that u u 0 0,D2 + hs D2 (u h, u h ) 1/2 C(h t+1 u t+1,d 2 + hs D 2 (u h, u h ) 1/2 + u u 0 1,D 2 ) C(h t+1 u t+1,d3 + h u u 0 0,D3 + u u 0 1,D3 + h 2 s D3 (u h, u h ) 1/2 ) Repeating this process l times, we have u u 0 0,D2 + hs D2 (u h, u h ) 1/2 C(h t+1 u t+1,d l + h l j u u 0 j,d + h l+1 s D (u h, u h ) 1/2 ) j=0 herefore, combining this estimate with (419), we derive u u h h,d0 C(h t u t+1,d + = C(h t u t+1,d + which completes the proof l h l j u u 0 j,d + h l+1 s D (u h, u h ) 1/2 ) j=0 l h l j u u 0 j,d + h l+1 s D (u u h, u u h ) 1/2 ), j=0 According to heorem 45, the interior energy error is bounded by the best local approximation in the WG finite element space h t u t+1,d, the interior negative

19 norms of the error l j=0 hl j u u 0 j,d, and the error on the element boundaries h l+1 s D (u u h, u u h ) 1/2 he last two components achieve the highest order of convergence when l = k 1 We here discuss the impact of the last two components on the interior energy convergence of the WG approximation Remark 46 aking l = k 1 in heorem 45, we observe that the error on the element boundaries h k s D (u u h, u u h ) 1/2 is of high order herefore, the interior energy error is in fact bounded by the best local approximation h t u t+1,d and the interior negative norms of the error k 1 j=0 hk 1 j u u 0 j,d Using the global negative norm estimate (39), for each interior negative norm, it holds that { h k 1 j u u 0 j,d Ch t +k u t +1,Ω if 0 j p 1, Ch t +k+p 1 j u t +1,Ω if j > p 1, where as defined in heorem 34, t = min(µ, k) with µ as the global regularity parameter for the solution u (namely, u H µ+1 (Ω)) and p = min(a, k) Recall the regularity index η > 1/2 (see (38)) for the dual problem (37) In the case η > k, we can take p = k such that the negative norms of the error are of higher order herefore, the interior energy norm of the error is bounded by the best local approximation error In the case η < k, the negative norms of the error can be the dominant terms in the interior energy error estimate For example, letting j = k 1 in the estimate above, we have u u 0 k+1,d Ch t +p u t +1,Ω, which may not be better than the best local approximation error his implies that the WG solution in the interior of the domain can be polluted by the singularities (associated to both the original and the dual problems) in other parts of the domain his phenomenon resembles the local behavior of the usual finite element method, which has been well documented in [19, 23, 26] 5 Numerical Results In this section, we report numerical results to illustrate the theoretical findings in the previous sections We implement the WG methods associated with the quasi-uniform triangulations h on two domains, the L-shaped domain and the square, to solve the Poisson equation 19 ( 1, 1) (1, 1) 3/4 (0, 0) (1, 0) 1/4 Ω D ( 1, 1) (0, 1) 1/4 3/4 Fig 51 Computational domains: the L-shaped domain Ω = ( 1, 1) 2 \ [(0, 1) ( 1, 0)] and the shaded interior region D = ( 1/4, 3/4) (1/4, 3/4) (left); the square domain Ω = (0, 1) 2 and the shaded interior region D = (1/4, 3/4) (1/4, 3/4) (right) 51 est 1 he first test is on the L-shaped domain Ω = ( 1, 1) 2 \ [(0, 1) ( 1, 0)] (Figure 51) Note that the regularity index η of the dual problem (37) (see also (38)) satisfies η = π/χ = 2/3 on Ω, where χ = 3π/2 is the largest opening angle associated to the vertex of the domain We choose the exact solution that possesses

20 20 the typical singularity near the reentrant corner at the origin u = r 2/3 sin( 2θ 3 ) r2 4, r = x 2 + y 2 herefore, u H 5/3 ɛ (Ω) for ɛ > 0 arbitrarily small Based on Proposition 22 and heorem 34, for any polynomial degree k 1, the global energy convergence rate of the WG solution is O(h 2/3 ɛ ), and the convergence rate in the global negative norm of order l 0 is O(h 4/3 ɛ ) In the interior region D = ( 1/4, 3/4) (1/4, 3/4), the solution u is smooth herefore, the estimate in heorem 45 (see also Remark 46) shows that the local energy convergence rate on D for the WG method should be O(h min(k,4/3 ɛ) ) instead of h k able 51 Error profiles for the WG methods on the L-shaped domain local error on D global error on Ω 1/h energy error rate energy error rate L 2 error rate k = E E E E E E E E E E E E E E E E E E k = E E E E E E E E E E E E E E E E E E k = E E E E E E E E E E E E E E E E E E In able 51 we display the convergence rates of the WG method with different polynomial degrees (k = 1, 2, 3) hese results include global convergence rates and the local convergence rates in the interior region D It is clear from this table that the global energy norm convergence is always comparable to O(h 2/3 ) and the global L 2 norm convergence behaves like O(h 4/3 ) In the interior region, for k = 1, the energy norm convergence is of first order O(h), which is optimal for linear approximations his is because, as mentioned above, the negative norms of the error in this case are of higher order O(h 4/3 ɛ ) and do not affect the local energy error For k = 2 and k = 3, based on our theory, the interior energy convergence rate shall be dominated by the convergence rate O(h 4/3 ɛ ) in negative norms his can

21 be observed in able 51 Hence, for k = 1, 2, 3, the numerical results are consistent with the theoretical predictions in heorem est 2 We also test the convergence of the WG methods with k = 1, 2 on the square domain Ω = (0, 1) 2 In this case, the regularity index η for the dual problem (37) is given by η = π/χ = 2, since the largest opening angle is π/2 We choose the exact solution as u = x(1 x)y(1 y)r 3/2 herefore, u H 3/2 ɛ (Ω) for ɛ > 0 arbitrarily small Note that the singularity in u is around the origin Based on Proposition 22, for the polynomial degree k = 1, 2, the global energy convergence rate of the WG solution is O(h 1/2 ɛ ) Meanwhile, because the dual problem is smoother (regularity index η = 2), according to heorem 34, when k = 1, 2, the convergence rate in the global L 2 norm shall be O(h 1/2+k ɛ ) In the interior region D = ( 1/4, 3/4) (1/4, 3/4), the solution u is smooth he estimate in heorem 45 shows that the local energy convergence rate on D for the WG method should be comparable to the best local approximation rate O(h k ) (k = 1, 2) able 52 Error profiles for the WG methods on the square domain local error on D global error on Ω 1/h energy error rate energy error rate L 2 error rate k = E E E E E E E E E E E E E E E E E E k = E E E E E E E E E E E E E E E E E E We list the numerical results of this test in able 52 It is clear from the table that the interior energy convergence rate is always optimal O(h k ) for k = 1, 2; and the convergence rates in global energy norm and in the global L 2 norm are O(h 1/2 ɛ ) and O(h 3/2 ɛ ), respectively his is in agreement with our theory developed in previous sections Appendix A he proof of Lemma 44

22 22 (A1) Proof It follows from (48), (27), and supp(ω 1 ) D 1, that a(ω 1 z, v) = a(z, ω 1 v) + (z 0 ω 1, v 0 ) + (z 0, (v 0 ω 1 )) + (l (z, v, ω 1 ) z 0, v 0 ω 1 n ) + ζ (z, v, ω 1 ) = I 1 + I 2 + I 3 + I 4 + I 5 Below, we obtain estimates for each I i, 1 i 5 Recall supp(ω 1 v) D 1 hen, using (211), (213), (214), (43), (33), and the inverse inequality, we have I 1 = a(z, ω 1 v) = a(q h u u h, ω 1 v Q h (ω 1 v)) + a(q h u u h, Q h (ω 1 v)) Q h u u h D2 ω 1 v Q h (ω 1 v) D2 + Ch t u t+1,d2 v D1 (A2) Ch z D2 v D2 + Ch t u t+1,d2 v D2 C(h t u t+1,d2 + z 0 0, D2 + hs D2 (z, z) 1/2 ) v D2 Using the boundedness of ω 1, (32), and the Cauchy-Schwarz inequality, we have (A3) and (A4) I 2 = (z 0 ω 1, v 0 ) C z 0, v D2 D2 I 3 = (z 0, (v 0 ω 1 )) C z 0, v D2 D2 Now, we estimate I 4 and I 5 Note that v b = 0 on D 2 herefore, hen, we have I 4 = z b, v b ω 1 n = 0 ( z 0 z b, Q h (ω 1 v 0 ) n ω 1 w v n z 0, v 0 ω 1 n ) = ( z 0 z b, (Q h (ω 1 v 0 ) ω 1 w v) n (A5) + z 0 z b, (Q h (v 0 ω 1 ) v 0 ω 1 ) n + z 0 z b, (v 0 v b ) ω 1 n v 0 v b, z 0 ω 1 n ) = A 1 + A 2 + A 3 A 4 By the Cauchy-Schwarz inequality, the trace inequality (31), the boundedness of ω 1,

23 23 (25), the inverse inequality, and (32), we have A 2 + A 3 (A6) Similarly, for A 4, we have A 4 C( (A7) z 0 z b 0, ( (Q h (v 0 ω 1 ) v 0 ω 1 ) n 0, + (v 0 v b ) ω 1 n 0, ) C( z 0 z b 2 0, ) 1/2 ( (Q h (v 0 ω 1 ) v 0 ω 1 ) n 2 0, + (v 0 v b ) ω 1 n 2 0, ) 1/2 C( z 0 z b 2 0, ) 1/2 ( (h 1 Q h (v 0 ω 1 ) v 0 ω 1 2 0, +h Q h (v 0 ω 1 ) v 0 ω 1 2 1, + v 0 v b 2 0, ) 1/2 C( z 0 z b 2 0, ) 1/2 ( h v 0 2 0, + v 0 v b 2 0, ) 1/2 Chs D2 (z, z) 1/2 v D2 v 0 v b 2 0, ) 1/2 ( h 1 z 0 ω 1 2 0, + h z 0 ω 1 2 1, ) 1/2 C( h 1 v 0 v b 0, ) 1/2 ( z 0 2 0, ) 1/2 C z 0 0, v D2 D2 Now, for A 1, we first have A 1 = z 0 z b, (Q h (ω 1 v 0 ) ω 1 w v) n = (A8) z 0 z b, Q h (ω 1 v 0 ω 1 w v) n + z 0 z b, (Q h (ω 1 w v) ω 1 w v) n = A 11 + A 12 For the second term A 12, using the Cauchy-Schwarz inequality, the trace inequality (31), the boundedness of ω 1, (25), the fact α ( w v) = 0 on for α = k, and the inverse inequality, we obtain A 12 z 0 z b 0, (Q h (ω 1 w v) ω 1 w v) n 0, C( z 0 z b 2 0, ) 1/2 ( h 1 Q h (ω 1 w v) ω 1 w v 2 0, (A9) +h Q h (ω 1 w v) ω 1 w v 2 1, ) 1/2 C( z 0 z b 2 0, ) 1/2 ( h 2k 1 w v 2 k 1, ) 1/2 C( z 0 z b 2 0, ) 1/2 ( h w v 2 0, ) 1/2 Chs D2 (z, z) 1/2 v D2

24 24 he estimate on A 11 is more involved Note that by (414), we have A 11 = z 0 z b, Q h (ω 1 v 0 ω 1 w v) n = ( z 0 w z, Q h (ω 1 v 0 ω 1 w v)) = (Q h (ω 1 z 0 ω 1 w z), v 0 w v) = v 0 v b, Q h (ω 1 z 0 ω 1 w z) n herefore, considering A 11 and I 5 together, by the Cauchy-Schwarz inequality, the trace inequality (31), the boundedness of ω 1, and (25), we obtain A 11 + I 5 = v 0 v b, Q h (ω 1 z 0 ω 1 w z) n + v 0 v b, (ω 1 w z Q h (ω 1 z 0 )) n + v 0 v b, Q h (z 0 ω 1 ) n = v 0 v b, (ω 1 w z Q h (ω 1 w z)) n + v 0 v b, Q h (z 0 ω 1 ) n D2 v 0 v b 0, ω 1 w z Q h (ω 1 w z) 0, + v 0 v b 0, Q h (z 0 ω 1 ) 0, C( v 0 v b 2 0, ) 1/2 ( h 1 ω 1 w z Q h (ω 1 w z) 2 0, +h ω 1 w z Q h (ω 1 w z) 2 1, ) 1/2 +( v 0 v b 2 0, ) 1/2 ( h 1 Q h (z 0 ω 1 ) 2 0, + h Q h (z 0 ω 1 ) 2 1, ) 1/2 C( v 0 v b 2 0, ) 1/2( ( h 1+2k ω 1 w z 2 k, ) 1/2 +( h 1 z 0 ω 1 2 0, + h z 0 ω 1 2 1, ) 1/2) hus, by the inverse inequality, (33), and the fact α ( w z) = 0 on for α = k, we have A 11 + I 5 C( v 0 v b 2 0, ) 1/2( ( h w z 2 0, ) 1/2 +( h 1 z 0 2 0, ) 1/2) (A10) C v D2 ( z 0 0, D2 + hs D2 (z, z) 1/2 ) It follows from the Cauchy-Schwarz inequality, the trace inequality, (23), (24),

25 25 and the inverse inequality that s D2 (z, z) = s (Q h u u h, z) = s (Q h u u, z) s (u h, z) C ( ( (h 2 u Q 0 u 0 2 0, + u Q 0 u 0 2 1, + h 1 u Q b u b 2 0, ) 1/2 +s D2 (u h, u h ) 1/2) s D2 (z, z) 1/2 C(h t u t+1,d2 + s D2 (u h, u h ) 1/2 )s D2 (z, z) 1/2, which implies (A11) s D2 (z, z) 1/2 C(h t u t+1,d2 + s D2 (u h, u h ) 1/2 ) In addition, the triangle inequality implies (A12) z 0 0, D2 Q h u 0 u 0, D2 + u u 0 0, D2 Ch t+1 u t+1,d2 + u u 0 0,D2 Combining all the estimates (A1) (A12), we have completed the proof for (415) REFERENCES [1] R Adams Sobolev Spaces, volume 65 of Pure and Applied Mathematics Academic Press, New York-London, 1975 [2] Apel and S Nicaise he finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges Math Methods Appl Sci, 21(6): , 1998 [3] C Băcuţă, V Nistor, and L Zikatanov Improving the rate of convergence of high order finite elements on polygons and domains with cusps Numer Math, 100(2): , 2005 [4] S C Brenner and L R Scott he mathematical theory of finite element methods, volume 15 of exts in Applied Mathematics Springer-Verlag, New York, second edition, 2002 [5] M Costabel and M Dauge Weighted regularization of maxwell equations in polyehdral domains a rehabilitation of nodal finite elements Numerische Mathematik, 93: , 2002 [6] M Costabel, M Dauge, and S Nicaise Corner singularities of Maxwell interface and eddy current problems In Operator theoretical methods and applications to mathematical physics, volume 147 of Oper heory Adv Appl, pages Birkhäuser, Basel, 2004 [7] M Costabel, M Dauge, and C Schwab Exponential convergence of hp-fem for Maxwell equations with weighted regularization in polygonal domains Math Models Methods Appl Sci, 15(4): , 2005 [8] M Dauge Elliptic Boundary Value Problems on Corner Domains, volume 1341 of Lecture Notes in Mathematics Springer-Verlag, Berlin, 1988 [9] P Grisvard Elliptic problems in nonsmooth domains, volume 24 of Monographs and Studies in Mathematics Pitman (Advanced Publishing Program), Boston, MA, 1985 [10] P Grisvard Singularities in boundary value problems, volume 22 of Research Notes in Applied Mathematics Springer-Verlag, New York, 1992 [11] B Guo and I Babuška Regularity of the solutions for elliptic problems on nonsmooth domains in R 3 I Countably normed spaces on polyhedral domains Proc Roy Soc Edinburgh Sect A, 127(1):77 126, 1997 [12] G Jin, H Li, Q Zhang, and Q Zou Linear and quadratic finite volume methods on triangular meshes for elliptic equations with singular solutions Int J Numer Anal Model, 13(2): , 2016 [13] V A Kondrat ev Boundary value problems for elliptic equations in domains with conical or angular points ransl Moscow Math Soc, 16: , 1967 [14] H Li he W 1 p stability of the Ritz projection on graded meshes Math Comp, 86(303):49 74, 2017