A dual singular complement method for the numerical solution of the Poisson equation with L 2 boundary data in non-convex domains
aa r X i v : . [ m a t h . NA ] M a y A dual singular complement methodfor the numerical solution of the Poisson equationwith L boundary data in non-convex domains ∗ Thomas Apel † Serge Nicaise ‡ Johannes Pfefferer § June 16, 2018
Abstract
The very weak solution of the Poisson equation with L boundary data isdefined by the method of transposition. The finite element solution with regularizedboundary data converges with order 1 / Key Words
Elliptic boundary value problem, very weak formulation, finite elementmethod, singular complement method, discretization error estimate
AMS subject classification
In this paper we consider the boundary value problem − ∆ y = f in Ω , y = u on Γ = ∂ Ω , (1.1)with right hand side f ∈ H − (Ω) and boundary data u ∈ L (Γ). We assume Ω ⊂ R to be a bounded polygonal domain with boundary Γ. Such problems arise in optimalcontrol when the Dirichlet boundary control is considered in L (Γ) only, see for examplethe papers by Deckelnick, G¨unther, and Hinze, [7], French and King, [8], and May,Rannacher, and Vexler, [11]. ∗ The work was partially supported by Deutsche Forschungsgemeinschaft, IGDK 1754. † [email protected] , Universit¨at der Bundeswehr M¨unchen, Institut f¨ur Mathematik und Bauin-formatik, D-85579 Neubiberg, Germany ‡ [email protected] , LAMAV, Institut des Sciences et Techniques de Valenciennes, Uni-versit´e de Valenciennes et du Hainaut Cambr´esis, B.P. 311, 59313 Valenciennes Cedex, France § [email protected] , Universit¨at der Bundeswehr M¨unchen, Institut f¨ur Mathematik undBauinformatik, D-85579 Neubiberg, Germany u ∈ L (Γ) we cannot expect a weak solution y ∈ H (Ω). Thereforewe define a very weak solution by the method of transposition which goes back at leastto Lions and Magenes [10]: Find y ∈ L (Ω) : ( y, ∆ v ) Ω = ( u, ∂ n v ) Γ − ( f, v ) Ω ∀ v ∈ V (1.2)with ( w, v ) G := R G wv denoting the L ( G ) scalar product or an appropriate dualityproduct. In our previous paper [1] we showed that the appropriate space V for the testfunctions is V := H (Ω) ∩ H (Ω) with H (Ω) := { v ∈ H (Ω) : ∆ v ∈ L (Ω) } . (1.3)In particular it ensures ∂ n v ∈ L (Γ) for v ∈ V such that the formulation (1.2) is welldefined. We proved the existence of a unique solution y ∈ L (Ω) for u ∈ L (Γ) and f ∈ H − (Ω), and that the solution is even in H / (Ω). The method of transposition isused in different variants also in [8, 2, 4, 3, 7, 11].Consider now the discretization of the boundary value problem. Let T h be a family ofquasi-uniform, conforming finite element meshes, and introduce the finite element spaces Y h = { v h ∈ H (Ω) : v h | T ∈ P ∀ T ∈ T h } , Y h = Y h ∩ H (Ω) , Y ∂h = Y h | ∂ Ω . Since the boundary datum u is in general not contained in Y ∂h we have to approximateit by L (Γ)-projection or by quasi-interpolation. We showed in [1] that we can constructin this way a function u h with k u − u h k H − / (Γ) ≤ Ch / k u k L (Γ) . As a side effect, the boundary datum is regularized since u h ∈ H / (Γ). Hence we canconsider a regularized (weak) solution in Y h ∗ := { v ∈ H (Ω) : v | Γ = u h } , y h ∈ Y h ∗ : ( ∇ y h , ∇ v ) Ω = ( f, v ) Ω ∀ v ∈ H (Ω) . (1.4)The finite element solution y h is now searched in Y ∗ h := Y h ∗ ∩ Y h and is defined in theclassical way: find y h ∈ Y ∗ h : ( ∇ y h , ∇ v h ) Ω = ( f, v h ) Ω ∀ v h ∈ Y h . (1.5)The same discretization was derived previously by Berggren [2] from a different point ofview. In [1] we showed that the discretization error estimate k y − y h k L (Ω) ≤ Ch s (cid:16) h / k f k H − (Ω) + k u k L (Γ) (cid:17) holds for s = 1 / y is alsoin H / (Ω) the convergence order is reduced; the finite element method does not leadto the best approximation in L (Ω). In order to describe the result we assume for2implicity that Ω has only one corner with interior angle ω ∈ ( π, π ). We proved in [1]the convergence order s ∈ (0 , λ − ), where λ := πω , and showed by numerical experimentsthat the order of almost λ − is sharp.In this paper, we modify the discrete solution y h from (1.5) in order to retain theconvergence order s = . In particular, we suggest to compute a function z h ∈ Y h ⊕ Span { r − λ sin( λθ ) } , where r, θ are polar coordinates at the concave corner, such that the error estimate k y − z h k L (Ω) ≤ Ch / (cid:16) h / k f k H − (Ω) + k u k L (Γ) (cid:17) can be shown. This method is a dual variant of the singular complement method intro-duced by Ciarlet and He [5]. Numerical experiments confirm the theoretical results. As in the introduction, let Ω be a domain with exactly one concave corner, and denotethis interior angle by ω ∈ ( π, π ). This corner is located at the origin of the coordinatesystem, and one boundary edge is contained in the positive x -axis. It is well knownthat the weak solution of the boundary value problem − ∆ v = g in Ω , v = 0 on Γ = ∂ Ω , (2.1)with g ∈ L (Ω) is not contained in H (Ω) but in H (Ω) ∩ H (Ω) = (cid:0) H (Ω) ∩ H (Ω) (cid:1) ⊕ Span { ξ ( r ) r λ sin( λθ ) } ,ξ being a cut-off function, see for example the monograph of Grisvard [9]. This meansthat R := { ∆ v : v ∈ H (Ω) ∩ H (Ω) } , is a closed subspace of L (Ω). It is shown in [9, Sect. 2.3] that L (Ω) = R ⊥ ⊕ Span { p s } , (2.2)with the dual singular function p s = r − λ sin( λθ ) + ˜ p s (2.3)where ˜ p s ∈ H (Ω) is chosen such that the decomposition (2.2) is orthogonal for the L (Ω) inner product. Therefore, the dual singular function p s is a solution of w ∈ L (Ω) : (∆ v, w ) = 0 ∀ v ∈ H (Ω) ∩ H (Ω) , (2.4)which proves the non-uniqueness of the solution of (2.4). This is the dual property tothe non-existence of a solution of (2.1) in H (Ω) ∩ H (Ω), see [9, Introduction].3ue to (2.2) we can split any L (Ω)-function into L (Ω)-orthogonal parts. To this enddenote by Π R and Π p s the orthogonal projections on R and on Span { p s } , respectively,i.e., for g ∈ L (Ω), it is g = Π R g + Π p s g whereΠ p s g = α ( g ) p s with α ( g ) = ( g, p s ) Ω k p s k L (Ω) , Π R g = g − Π p s g. Since p s ∈ L (Ω) there exists φ s ∈ H (Ω) ∩ H (Ω) : − ∆ φ s = p s , (2.5)see also Section 4 for more details on φ s . For the moment we assume that p s and φ s areexplicitly known; hence the decomposition g = Π R g + α ( g ) p s can be computed once g is given. Computable approximations of p s and φ s are discussed in Section 4.Now we come back to problem (1.2) and decompose its solution y in the form y = Π R y + α ( y ) p s . (2.6)From the decomposition (2.2) we see that problem (1.2) is equivalent to( y, p s ) Ω = − ( u, ∂ n φ s ) Γ + ( f, φ s ) Ω , ( y, ∆ v ) Ω = ( u, ∂ n v ) Γ − ( f, v ) Ω ∀ v ∈ H (Ω) ∩ H (Ω)and with the orthogonal splitting (2.6) to α ( y ) ( p s , p s ) Ω = − ( u, ∂ n φ s ) Γ + ( f, φ s ) Ω , (Π R y, ∆ v ) Ω = ( u, ∂ n v ) Γ − ( f, v ) Ω ∀ v ∈ H (Ω) ∩ H (Ω) . The first equation directly yields α ( y ), namely α ( y ) = − ( u, ∂ n φ s ) Γ + ( f, φ s ) Ω ( p s , p s ) Ω , (2.7)hence the projection of y on p s is known. It remains to find an approximation of Π R y .At this point we recall the regularization approach from [1] which we summarizedalready in the introduction. Let u h ∈ H / (Γ) be a regularized boundary datum suchthat we can define the regularized (weak) solution in Y h ∗ := { v ∈ H (Ω) : v | Γ = u h } , y h ∈ Y h ∗ : ( ∇ y h , ∇ v ) Ω = ( f, v ) Ω ∀ v ∈ H (Ω) . (2.8)In [1] we showed that the regularization error can be estimated by k y − y h k L (Ω) ≤ c k u − u h k H − s (Γ) where s = if Ω is convex and s ∈ [0 , λ − ) if Ω is non-convex, that means theregularization error is in general bigger in the non-convex case. With the next lemmawe show that Π R ( y − y h ) is not affected by non-convex corners.4 emma 1. If the domain Ω is non-convex, the estimate k Π R ( y − y h ) k L (Ω) ≤ C k u − u h k H − / (Γ) holds.Proof. Recall V = H (Ω) ∩ H (Ω) from (1.3). From (2.8) and the Green formula, wehave for any v ∈ V ( f, v ) Ω = ( ∇ y h , ∇ v ) Ω = − ( y h , ∆ v ) Ω + ( y h , ∂ n v ) Γ . Note that v ∈ V is sufficient, see [6, Lemma 3.4]. Subtracting this expression from thevery weak formulation (1.2), we get( y − y h , ∆ v ) Ω = ( u − u h , ∂ n v ) Γ ∀ v ∈ V. Restricting this identity to v ∈ H (Ω) ∩ H (Ω), we have(Π R ( y − y h ) , ∆ v ) Ω = ( u − u h , ∂ n v ) Γ ∀ v ∈ H (Ω) ∩ H (Ω) . (2.9)Now for any z ∈ R , we let v z ∈ H (Ω) ∩ H (Ω) be the unique solution of∆ v z = z, (2.10)that satisfies k ∂ n v z k H / (Γ) ≤ c k v z k H (Ω) ≤ c k z k L (Ω) . (2.11)Since for any g ∈ L (Ω) the equality(Π R ( y − y h ) , g ) Ω = (Π R ( y − y h ) , Π R g ) Ω = ( y − y h , Π R g ) Ω holds we get with (2.9)–(2.11) k Π R ( y − y h ) k L (Ω) = sup z ∈ R,z =0 ( y − y h , z ) Ω k z k L (Ω) = sup z ∈ R,z =0 ( u − u h , ∂ n v z ) Γ k z k L (Ω) ≤ k u − u h k H − / (Γ) sup z ∈ R,z =0 k ∂ n v z k H / (Γ) k z k L (Ω) ≤ c k u − u h k H − / (Γ) which is the estimate to be proved. Recall from the introduction the finite element spaces Y h = { v h ∈ H (Ω) : v h | T ∈ P ∀ T ∈ T h } , Y h = Y h ∩ H (Ω) , Y ∂h = Y h | ∂ Ω , T h of quasi-uniform, conforming finite element meshes. Assume thatthe regularized boundary datum u h is contained in Y ∂h such that the estimates k u h k L (Γ) ≤ c k u k L (Γ) , (3.1) k u − u h k H − / (Γ) ≤ Ch / k u k L (Γ) , (3.2)hold. It is proved in [1] that this can be accomplished by using the L (Γ)-projection orby quasi-interpolation. A consequence of Lemma 1 is the estimate k Π R ( y − y h ) k L (Ω) ≤ Ch / k u k L (Γ) (3.3)in the case of a non-convex domain Ω. (In the case of a convex domain the operator Π R is the identity, and the corresponding error estimates were already proven in [1].)As already done in the introduction, define further the finite element solution y h ∈ Y ∗ h := Y h ∗ ∩ Y h via y h ∈ Y ∗ h : ( ∇ y h , ∇ v h ) Ω = ( f, v h ) Ω ∀ v h ∈ Y h . (3.4)We proved in [1] that k y − y h k L (Ω) ≤ Ch s (cid:16) h / k f k H − (Ω) + k u k L (Γ) (cid:17) (3.5)holds for s = if the domain is convex but only s ∈ (0 , λ − ) in the non-convex case.In the next lemma we show that Π R ( y − y h ) is not affected by the non-convex corners. Lemma 2.
For non-convex domains Ω the discretization error estimate k Π R ( y − y h ) k L (Ω) ≤ Ch / (cid:16) h / k f k H − (Ω) + k u k L (Γ) (cid:17) holds.Proof. By the triangle inequality we have k Π R ( y − y h ) k L (Ω) ≤ k Π R ( y − y h ) k L (Ω) + k Π R ( y h − y h ) k L (Ω) . (3.6)The first term is estimated in (3.3). For the second term we first notice that y h − y h ∈ H (Ω) satisfies the Galerkin orthogonality( ∇ ( y h − y h ) , ∇ v h ) Ω = 0 ∀ v h ∈ Y h , (3.7)see (1.4) and (1.5). With that, we estimate k Π R ( y h − y h ) k L (Ω) by a similar argumentsas k Π R ( y − y h ) k L (Ω) in the proof of Lemma 1. Recall from (2.10) and (2.11) that v z ∈ H (Ω) ∩ H (Ω) is the weak solution of ∆ v z = z ∈ R . It can be approximated bythe Lagrange interpolant I h v z satisfying k∇ ( v z − I h v z ) k L (Ω) ≤ ch k v z k H (Ω) ≤ ch k z k L (Ω) .
6e get k Π R ( y h − y h ) k L (Ω) = sup z ∈ R,z =0 ( y h − y h , z ) Ω k z k L (Ω) = sup z ∈ R,z =0 ( ∇ ( y h − y h ) , ∇ v z ) Ω k z k L (Ω) = sup z ∈ R,z =0 ( ∇ ( y h − y h ) , ∇ ( v z − I h v z )) Ω k z k L (Ω) ≤ ch k∇ ( y h − y h ) k L (Ω) . (3.8)In order to bound k∇ ( y h − y h ) k L (Ω) by the data we consider a lifting B h u h ∈ Y ∗ h defined by the nodal values as follows:( B h u h )( x ) = ( u h ( x ) , for all nodes x ∈ Γ , x ∈ Ω . (3.9)The homogenized solution y h = y h − B h u h ∈ H (Ω) satisfies( ∇ y h , ∇ v ) Ω = ( f, v ) Ω − ( ∇ ( B h u h ) , ∇ v ) Ω ∀ v ∈ H (Ω) . By taking v = y h we see that k∇ y h k L (Ω) ≤ k f k H − (Ω) k y h k H (Ω) + k∇ ( B h u h ) k L (Ω) k∇ y h k L (Ω) . Using the Poincar´e inequality we obtain k∇ y h k L (Ω) ≤ c k f k H − (Ω) + k∇ ( B h u h ) k L (Ω) , (3.10)and with the C´ea lemma k∇ ( y h − y h ) k L (Ω) ≤ k∇ ( y h − B h u h ) k L (Ω) = k∇ y h k L (Ω) ≤ c k f k H − (Ω) + k∇ ( B h u h ) k L (Ω) . The remaining term k∇ ( B h u h ) k L (Ω) is estimated by using the inverse inequality k∇ ( B h u h ) k L ( T ) ≤ ch − / k u h k L ( E ) . for E ⊂ T ∩ Γ, T ∈ T h , which can be proved by standard scaling arguments, to get k∇ ( B h u h ) k L (Ω) ≤ ch − / k u h k L (Γ) . (3.11)Hence we proved k∇ ( y h − y h ) k L (Ω) ≤ c k f k H − (Ω) + ch − / k u h k L (Γ) . With (3.6), (3.3), (3.8), the previous inequality, and (3.1) we finish the proof.With (2.6) we can immediately conclude the following result.7 orollary 3.
Let Ω be a non-convex domain and let y h ∈ Y ∗ h be the solution of (3.4) ,then the discretization error estimate k y − (Π R y h + α ( y ) p s ) k L (Ω) ≤ Ch / (cid:16) h / k f k H − (Ω) + k u k L (Γ) (cid:17) holds, reminding that p s and α ( y ) are given by (2.3) and (2.7) , respectively. Hence the positive result is that Π R y h + α ( y ) p s is a better approximation of y than y h .The problem is that p s and φ s are used explicitly, and in practice they are not known.A remedy of this drawback is the aim of the next section. Following [5], we approximate p s from (2.3) by p hs = p ∗ h − r h + r − λ sin( λθ ) , r h = B h (cid:16) r − λ sin( λθ ) (cid:17) ,p ∗ h ∈ Y h : ( ∇ p ∗ h , ∇ v h ) Ω = ( ∇ r h , ∇ v h ) Ω ∀ v h ∈ Y h , (4.1)with B h from (3.9). The function φ s from (2.5) admits the splitting φ s = ˜ φ + βr λ sin( λθ ) , (4.2)with ˜ φ ∈ H (Ω) and β = π − k p s k L (Ω) , see again [5]. It is approximated by φ hs = φ ∗ h − β h s h + β h r λ sin( λθ ) , s h = B h (cid:16) r λ sin( λθ ) (cid:17) , β h = 1 π k p hs k L (Ω) ,φ ∗ h ∈ Y h : ( ∇ φ ∗ h , ∇ v h ) Ω = ( p hs , v h ) Ω + β h ( ∇ s h , ∇ v h ) Ω ∀ v h ∈ Y h , (4.3)that means, ˜ φ is approximated by ˜ φ h = φ ∗ h − β h s h ∈ Y h . The approximation errors arebounded by k p s − p hs k L (Ω) ≤ ch λ − ǫ ≤ ch, (4.4) | β − β h | ≤ ch λ − ε ≤ ch, (4.5) k φ s − φ hs k , Ω ≤ ch, (4.6)see [5, Lemmas 3.1–3.3], where (4.5) and (4.6) imply k ˜ φ − ˜ φ h k , Ω ≤ ch. (4.7)At the end of Section 3 we saw that Π R y h + α ( y ) p s is a better approximation of y than y h . Since this function is not computable we approximate it by z h = Π hR y h + α h p hs , (4.8)8ith Π hR y h = y h − γ h p hs , γ h = ( y h , p hs ) Ω k p hs k L (Ω) (4.9)and a suitable approximation α h of α ( y ) = − ( u, ∂ n φ s ) Γ + ( f, φ s ) Ω ( p s , p s ) Ω from (2.7). To this end we write the problematic term by using (4.2) as( u, ∂ n φ s ) Γ = ( u, ∂ n ˜ φ ) Γ + β ( u, ∂ n ( r λ sin( λθ ))) Γ . and replace the term ( u, ∂ n ˜ φ ) Γ by ( u h , ∂ n ˜ φ ) Γ . Since ˜ φ belongs to H (Ω) and u h is thetrace of B h u h , we get by using the Green formula( u h , ∂ n ˜ φ ) Γ = ( B h u h , ∆ ˜ φ ) Ω + ( ∇ B h u h , ∇ ˜ φ ) Ω = − ( B h u h , p s ) Ω + ( ∇ B h u h , ∇ ˜ φ ) Ω (4.10)as ∆ ˜ φ = ∆ φ s = − p s . With all these notations and results, we define α h = ( B h u h , p hs ) Ω − ( ∇ B h u h , ∇ ˜ φ h ) Ω − β h ( u, ∂ n ( r λ sin( λθ ))) Γ + ( f, φ hs ) Ω ( p hs , p hs ) . (4.11)Note that α h can be computed explicitly and therefore z h as well.Let us estimate the approximation errors made. Lemma 4.
Let Ω be a non-convex domain and let y h ∈ Y ∗ h be the solution of (3.4) .Then the error estimates k Π R y h − Π hR y h k L (Ω) ≤ ch (cid:0) k f k H − (Ω) + k u k L (Γ) (cid:1) , (4.12) | α ( y ) − α h | ≤ ch / (cid:16) h / k f k H − (Ω) + k u k L (Γ) (cid:17) (4.13) hold.Proof. With the definitions of Π R and Π hR , with γ := ( y h , p s ) Ω / k p s k L (Ω) , and by usingthe triangle inequality we have k Π R y h − Π hR y h k L (Ω) = k γp s − γ h p hs k L (Ω) ≤ | γ − γ h | k p hs k L (Ω) + | γ | k p s − p hs k L (Ω) We write γ − γ h = ( y h , p s ) Ω k p s k L (Ω) − ( y h , p hs ) Ω k p hs k L (Ω) = ( y h , p s − p hs ) Ω k p s k L (Ω) + ( y h , p hs ) Ω k p s k L (Ω) − k p hs k L (Ω) ! = ( y h , p s − p hs ) Ω k p s k L (Ω) + ( y h , p hs ) Ω ( p hs + p s , p hs − p s ) Ω k p s k L (Ω) k p hs k L (Ω) , | γ − γ h | ≤ ch k y h k L (Ω) . We have used that k p s k L (Ω) and k p hs k L (Ω) can be treated as constants due to thedefinition of p s and due to (4.4). We conclude with | γ | ≤ c k y h k L (Ω) , and (4.4) that k Π R y h − Π hR y h k L (Ω) ≤ ch k y h k L (Ω) . (4.14)In view of the finite element error estimate (3.5) and the standard a priori estimate forthe very weak solution, k y k L (Ω) ≤ c (cid:0) k f k H − (Ω) + k u k L (Γ) (cid:1) , see Lemma 2.3 of [1], we have k y h k L (Ω) ≤ k y k L (Ω) + k y − y h k L (Ω) ≤ c (cid:0) k f k H − (Ω) + k u k L (Γ) (cid:1) . This estimate together with (4.14) proves (4.12).The proof of the estimate (4.13) is based on writing the problematic term in thedefinition of α ( y ) without approximation as( u, ∂ n φ s ) Γ = ( u, ∂ n ˜ φ ) Γ + β ( u, ∂ n ( r λ sin( λθ ))) Γ = ( u − u h , ∂ n ˜ φ ) Γ + ( u h , ∂ n ˜ φ ) Γ + β ( u, ∂ n ( r λ sin( λθ ))) Γ = ( u − u h , ∂ n ˜ φ ) Γ − ( B h u h , p s ) Ω + ( ∇ B h u h , ∇ ˜ φ ) Ω + β ( u, ∂ n ( r λ sin( λθ ))) Γ where we used (4.10) in the last step. Consequently, we showed that α ( y ) − α h = 1 k p s k L (Ω) (cid:16) − ( u − u h , ∂ n ˜ φ ) Γ + ( B h u h , p s − p hs ) Ω − ( ∇ B h u h , ∇ ( ˜ φ − ˜ φ h )) Ω − ( β − β h ) ( u, ∂ n ( r λ sin( λθ ))) Γ + ( f, φ s − φ hs ) Ω (cid:17) . To prove (4.13), in view of (4.4), (4.5), and (4.6) it remains to show that (cid:12)(cid:12)(cid:12) ( u − u h , ∂ n ˜ φ ) Γ (cid:12)(cid:12)(cid:12) ≤ ch / k u k L (Γ) , (cid:12)(cid:12)(cid:12) ( B h u h , p s − p hs ) Ω (cid:12)(cid:12)(cid:12) ≤ ch / k u k L (Γ) , (cid:12)(cid:12)(cid:12) ( ∇ B h u h , ∇ ( ˜ φ − ˜ φ h )) Ω (cid:12)(cid:12)(cid:12) ≤ ch / k u k L (Γ) . The first estimate follows from the estimate (3.2) and the fact that ˜ φ belongs to H (Ω).The second one follows from the Cauchy-Schwarz inequality and the estimates (3.11)and (4.4). Similarly, the third estimate follows from the Cauchy-Schwarz inequality andthe estimates (3.11) and (4.7). 10 orollary 5. Let Ω be a non-convex domain and let y h ∈ Y ∗ h be the solution of (3.4) and let z h be derived by (4.8) , (4.9) , and (4.11) , then the discretization error estimate k y − z h k L (Ω) ≤ Ch / (cid:16) h / k f k H − (Ω) + k u k L (Γ) (cid:17) holds.Proof. The main ingredients of the proof were already derived. Indeed, it is k y − z h k L (Ω) = k Π R y + α ( y ) p s − Π hR y h − α h p hs k L (Ω) ≤ k Π R y − Π R y h k L (Ω) + k Π R y h − Π hR y h k L (Ω) + | α ( y ) − α h | k p s k L (Ω) + | α h | k p s − p hs k L (Ω) . The first three terms can be estimated by using Lemmas 2 and 4. So it remains totreat the fourth term. To bound | α h | we use the triangle inequality | α h | ≤ | α h − α ( y ) | + | α ( y ) | . For the first term we use (4.13), while for the second term we use (2.7) reminding that φ s belongs to H / ǫ (Ω) with some ǫ >
0. Altogether we have | α h | ≤ C (cid:0) k f k H − (Ω) + k u k L (Γ) (cid:1) and conclude by using (4.4).Before we describe the numerical experiments, let us summarize the algorithm.1. Compute the finite element solution y h ∈ Y ∗ h : ( ∇ y h , ∇ v h ) Ω = ( f, v h ) Ω ∀ v h ∈ Y h where Y ∗ h = { v h ∈ Y h : v h | Γ = u h } , compare (1.5), with u h ∈ Y ∂h being anapproximation of the boundary datum u satisfying (3.1) and (3.2).2. Compute the approximate singular functions: r h = B h (cid:16) r − λ sin( λθ ) (cid:17) ,p ∗ h ∈ Y h : ( ∇ p ∗ h , ∇ v h ) Ω = ( ∇ r h , ∇ v h ) Ω ∀ v h ∈ Y h , ˜ p h = p ∗ h − r h ,β h = 1 π k ˜ p h + r − λ sin( λθ ) k L (Ω) ,s h = B h (cid:16) r λ sin( λθ ) (cid:17) ,φ ∗ h ∈ Y h : ( ∇ φ ∗ h , ∇ v h ) Ω = (˜ p h + r − λ sin( λθ ) , v h ) Ω + β h ( ∇ s h , ∇ v h ) Ω ∀ v h ∈ Y h , ˜ φ h = φ ∗ h − β h s h , compare (4.1) and (4.3). 11. Compute γ h = ( y h , p hs ) Ω ( p hs , p hs ) Ω with p hs = ˜ p h + r − λ sin( λθ ) ,α h = ( B h u h , p hs ) Ω − ( ∇ B h u h , ∇ ˜ φ h ) Ω − β h ( u, ∂ n ( r λ sin( λθ ))) Γ + ( f, φ hs ) Ω ( p hs , p hs ) ,δ h = α h − γ h , ˜ z h = y h + δ h ˜ p h , compare (4.9) and (4.11). According to (4.8), the numerical solution is z h = ˜ z h + δ h r − λ sin( λθ ) . Note that all integrals with r λ and r − λ must be computed with care. This section is devoted to the numerical verification of our theoretical results. Forthat purpose we present examples with known solution. Furthermore, to examine theinfluence of the corner singularities, we consider several polygonal domain Ω ω dependingon an interior angle ω ∈ (0 , π ). The computational domains are defined byΩ ω := ( − , ∩ { x ∈ R : ( r ( x ) , θ ( x )) ∈ (0 , √ × [0 , ω ] } , (5.1)where r and θ stand for the polar coordinates located at the origin. The boundary ofΩ ω is denoted by Γ ω . We solve the problem − ∆ y = 0 in Ω ω , y = u on Γ , (5.2)numerically by using the proposed dual singular function method. The boundary datum u is chosen as follows u := r − . sin( − . θ ) on Γ ω . This function belongs to L p (Γ) for every p < . y = r − . sin( − . θ ) , since y is harmonic.The quasi-uniform finite element meshes for the calculations are generated by using anewest vertex bisection algorithm. The discretization errors for different mesh sizes andthe corresponding experimental orders of convergence are given in Table 1 for differentinterior angles ω = 270 ◦ and ω = 355 ◦ . We see that the numerical results confirm theexpected convergence rate 1 / u, ∂ n ( r λ sin( λθ ))) Γ h k e h k L (Ω ω ) eoc0.25000 0.587250.12500 0.42338 0.472010.06250 0.30318 0.481770.03125 0.21606 0.488700.01562 0.15352 0.493020.00781 0.10888 0.495720.00390 0.07712 0.49742 mesh size h k e h k L (Ω ω ) eoc0.25000 1.020690.12500 0.83402 0.291390.06250 0.58964 0.500250.03125 0.41696 0.499910.01562 0.29506 0.498900.00781 0.20903 0.497250.00390 0.14836 0.49462Table 1: Discretization errors e h = y − z h for ω = 3 π/ ω = 355 π/
180 (right)has to be adapted in order to get a sufficiently good approximation. Otherwise, the errordue to the quadrature formula dominates the overall error. In our implementation, wechose for the numerical integration a graded mesh on the boundary ( h E ∼ hr − µE if thedistance r E of the boundary edge E satisfies 0 < r E < R with R being the radius ofthe refinement zone and µ being the refinement parameter, and h T = h /µ for r E = 0)combined with a one-point Gauss quadrature rule on each element. Furthermore, thegrading parameter µ is chosen such that µ ≤ π/ω − , which seems to be the correct grading to achieve a convergence order of 1 /
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