POISSON EQUATIONS JEONGHUN LEE (1)

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1 POISSON EQUATIONS JEONGHUN LEE Throughout this note the symbol X À Y stands for an inequality X CY with some constant C 0 which is independent of mesh sizes h. However, we will use X CY when we want to emphasize the constant C 0 in the context. 1. Mixed form of the Poisson equation Let Ω R n, n 2, 3 be a bounded domain with Lipschitz boundary. For a given function f : Ω Ñ R, Poisson equation with homogeneous Dirichlet boundary condition is to find a function u : Ω Ñ R such that (1) u f in Ω, u 0 on Γ : BΩ. Since we will not discuss regularity theory of PDEs in this note, we assume that BΩ is at least Lipschitz continuous. For f P H 1 pωq a variational formulation of (1) is to find u P H0 1 pωq such that (2) p u, vq pf, vq, v P H 1 0 pωq. One can prove that this problem is well-posed using Poincare inequality and the Lax Milgram lemma. Furthermore, there is a constant C 0 depending only on Ω such that (3) }u} 1 C}f} 1. In mixed formulation of Poisson problems, we introduce σ : u as a new variable and (1) is rewritten as (4) σ u, div σ f in Ω, u 0 on Γ. Let Hpdiv, Ωq be the subspace of L 2 pω; R n q such that their divergence is also square integrable. For τ P Hpdiv, Ωq the } } div norm is defined by }τ} 2 div }τ}2 0 } div τ} 2 0. Now we derive a variational formulation of (4). By applying the integration by parts to the first equation in (4) with u 0 on Γ, we have pσ, τq p u, τq I.B.P pu, div τq, τ P Hpdiv, Ωq. 1

2 2 JEONGHUN LEE Thus a variational formulation of (4) is: find pσ, uq P Hpdiv, Ωq L 2 pωq such that (5) (6) pσ, τq pu, div τq 0, τ P Hpdiv, Ωq, pdiv σ, vq pf, vq, In the context of abstract formulation, Σ Hpdiv, Ωq, apσ, τq pσ, τq, V L 2 pωq, v P L 2 pωq. bpτ, vq pdiv τ, vq. Let us show that (5 6) is well-posed for f P L 2 pωq V. By the Brezzi s theorem it suffices to check the first and second LBB conditions. Here we will check the coercivity condition and the inf-sup condition. Regarding the bilinear form bpσ, vq pdiv σ, vq, one sees that the space Z is Z tτ P Hpdiv, Ωq div τ 0u. The Z-coercivity condition (or LBB1 condition) can be checked easily by apτ, τq }τ} 2 0 }τ} 2 div, τ P Z. To see the inf-sup condition (or LBB2 condition) we need a preliminary result. Lemma 1. Let Ω be a bounded domain in R n, n 2, 3, and v P L 2 pωq. Then there exists ξ P H 1 pω; R n q such that div τ v in Ω and with C 0 depending only on Ω. }ξ} 1 C}v} 0, Proof. Let B be a ball containing Ω with the smallest radius and define ṽ P L 2 pbq by # vpxq, if x P Ω, ṽpxq 0, if x R Ω. It is obvious that }v} 0,Ω }ṽ} 0,B. By a well-known theory of elliptic PDEs, there exists a function w P H 2 pbq such that w ṽ in B, w 0 on BB, }w} 2,B C}ṽ} 0,B, with some C 0. Take ξ as the restriction of w on Ω. Then div ξ div w v in Ω, and }ξ} 1,Ω } w} 1,B }w} 2,B C}ṽ} 0,B C}v} 0,Ω. Proof is completed. Remark 1. Here we note that the values of ξ on BΩ are not specified.

3 POISSON EQUATIONS 3 To check the inf-sup condition, we prove the following claim implying the inf-sup condition, which is, for any v P L 2 pωq there exists τ P Hpdiv, Ωq such that div τ v and }τ} div c}v} 0. To prove it, note that, for given v P L 2 pωq there exists τ P H 1 pω; R n q such that div τ v, }τ} 1 c}v} 0 by the above Lemma. Since }τ} div }τ} 1, the claim follows. 2. Stable finite elements In this section we introduce stable mixed finite elements for the Poisson problems. Throughout this note, T h will denote the set of triangles/tetrahedra in a triangulation of Ω. However, we will also use T h to denote the mesh given by the triangulation. We denote the set of edges/faces by E h and E h te P E h E Ωu, E B h te P E h E Γu. We use P k pt q to denote the space of polynomials on T P T h of degree k for an integer k 0. The R n -valued polynomial space pp k pt qq n is denoted by P k pt ; R n q. We define P k pt h q tv P L 2 pωq v T P P k pt q, T P T h u, and P k pt h ; R n q is defined in a similar way. For the space of homogeneous polynomials of degree k we use the symbol H k. All variations of polynomial spaces are similarly defined. In order to find a stable mixed finite elements for the Poisson problems we need to find two finite element spaces Σ h Hpdiv, Ωq, V h L 2 pωq satisfying the (discrete) LBB conditions. For τ P P k pt h ; R n q it is known that τ P Hpdiv, Ωq if and only if the normal components of τ is continuous on every E P E h. The following theorem gives a mathematical statement of this claim. Theorem 1. Suppose that τ P P k pt h ; R n q. For and edge/face E P E h, let T E, T E be the two distinct triangles/tetrahedra sharing E as the common boundary and n E, n E be the unit normal vectors on E coming out from T E, T E, respectively. We use τ n E to denote the inner product of the trace of τ from the interior of T E, and n E. In a similar way, we define τ n E with T E, n E. Then τ P Hpdiv, Ωq if and only if τ n E τ n E for every E P E h. Exercise. Recall that τ P Hpdiv, Ωq if there exists v P L 2 pωq such that pτ, φq pv, P C 8 0 pωq, where C0 8pΩq tφ P C8 pωq supp φ Ωu. Prove the above theorem (Hint : the integration by parts). In two and three dimensions, x will denote the column vectors x 1 x1 x, x. x 2 x 2 x 3

4 4 JEONGHUN LEE 2.1. Two families of mixed finite elements. In this section we introduce two well-known families of mixed finite elements for the Poisson problems. We will show precise definition of those elements only for the lowest order case in this section. Complete definitions of higher order elements and investigation of their properties will be discussed later. In order to define a finite element space Σ h we need the space of shape functions Σ T on T and local degrees of freedom (DOF). Definition 1 (Raviart Thomas Nedelec (RTN) space). For T P T h and k 0, the space of shape functions Σ T is (7) Σ T P k pt ; R n q x P k p ˆT q rp k p ˆT qs n ` x H k p ˆT q. The DOFs for k 0 is given by τ P Σ T ÞÑ τ n E ds, E E BT. Definition 2 (Brezzi Douglas Marini (BDM) space). For a triangle/tetrahedron T and k 0, the space of shape functions Σ T is (8) Σ T P k 1 pt ; R n q. The DOFs for k 0 are τ P Σ T ÞÑ E τ n E q ds, q P P 1 peq, E P BT. These shape functions and local DOFs give consistent global DOFs to define a finite element space Σ h on T h. We have not defined V h space yet. For V h we choose (9) V h tv P L 2 pωq v T P P k pt q, T P T h u. From now on, when we say pσ h, V h q, Σ h is a RTN or BDM space with shape functions in (7) or (8), and V h is the space in (9) with same k 0. One can see that div Σ h V h by checking degrees of piecewise polynomials. Let P h : L 2 pωq Ñ V h be the orthogonal L 2 projection. Now we define interpolation operators mapping H 1 pω; R n q into Σ h and introduce useful properties of them. In this section we do not show full proofs of those properties because we will discuss them later in more detail. For τ P H 1 pt ; R n q, the map Π T : H 1 pt ; R n q Ñ Σ T is defined by Π T τ n E ds τ n E ds, E BT, prtnq E E Π T τ n E q ds τ n E q ds, E BT, q P P 1 peq. pbdmq. E E From this local interpolation operator we can define Π h : H 1 pω; R n q Ñ Σ h such that Π h T Π T for T P T h. Here we summarize some properties of Π h.

5 For τ P H 1 pω; R n q, (10) (11) (12) POISSON EQUATIONS 5 }Π h τ} 0 c}τ} 1, div Π h τ P h div τ, # ch k 1 }τ} k 1, }τ π h τ} 0 ch k 2 }τ} k 2, prtnq pbdmq Remark 2. The operator Π h is not well-defined on Hpdiv, Ωq because the trace of normal component of Hpdiv, T q function belongs to H 1{2 pbt q, so the integration of normal component of τ on E BT is not well-defined Stability of RTN and BDM elements. Theorem 2. The pair pσ h, V h q is a stable mixed finite element for the Poisson problems. Proof. Since div Σ h V h, Z h tτ P Σ h pdiv τ, vq P V h u tτ P Σ h div τ 0u. The LBB1 condition holds because pτ, τq }τ} 2 0 }τ} 2 div, τ P Z h. For the inf-sup (LBB2) condition, recall that for any v P V h there exists ξ P H 1 pω; R n q such that div ξ v and }ξ} 1 c}v} 0. If we set τ Π h ξ, then and div τ div Π h ξ P h div ξ P h v v, }τ} div cp}τ} 0 } div τ} 0 q cp}π h ξ} 0 }v} 0 q Thus the inf-sup condition holds. p7 v P V h q cp}ξ} 1 }v} 0 q p7 }Π h ξ} 0 c}ξ} 1 q c}v} 0 p7 }ξ} 1 c}v} 0 q Remark 3. As a corollary of the inf-sup condition proof, one can see that div Σ h V h. 3. A priori error analysis 3.1. Improved error estimates. For stable finite elements pσ h, V h q the discrete problem (13) (14) pσ h, τq pu h, div τq 0, τ P Σ h, pdiv σ h, vq pf, vq, v P V h, has a unique solution. Let pσ h, u h q be the solution of the above problem. An immediate consequence of abstract theory of saddle point problems gives }σ σ h } div }u u h } 0 À inf pτ,vqpσ h V h p}σ τ} div }u v} 0 q.

6 6 JEONGHUN LEE By the approximability of BDM k and RTN k, }σ σ h } div }u u h } 0 À h m p} div σ} m }u} m q, 1 m k. Note that this is not fully satisfactory because the full approximability of Σ h in BDM k is not used. Thus we will show an improved error estimate of }σ σ h } 0. The difference of (5 6) and (13 14), gives (15) (16) pσ σ h, τq pu u h, div τq 0, τ P Σ h, pdivpσ σ h q, vq 0, v P V h. Since pdiv σ, vq pp h div σ, vq for v P V h, (16) implies that div σ h P h div σ. If we take τ Π h σ σ h in (15), then because divpπ h σ σ h q 0. Then pσ σ h, Π h σ σ h q 0, }Π h σ σ h } 2 0 pπ h σ σ, Π h σ σ h q }Π h σ σ} 0 }Π h σ σ h } 0, so }Π h σ σ h } 0 }σ Π h σ} 0. By the triangle inequality, }σ σ h } 0 }σ Π h σ} 0 }Π h σ σ h } 0 2}σ Π h σ} 0 # À h m }σ} m, 1 m k 1, pbdmq 1 m k, prtnq Superconvergence of }P h u u h } 0. Now we show a superconvergence result of }P h u u h } 0. Note that the orthogonality pu P h u, div τq P Σ h, allows us to reduce (15) to (17) pσ σ h, τq pp h u u h, div τq 0, τ P Σ h. By the inf-sup condition, there exists τ P Σ h such that div τ P h u u h and }τ} div C}P h u u h } 0. With such τ, we obtain therefore }P h u u h } 2 0 pσ σ h, τq À }σ σ h } 0 }P h u u h } 0, }P h u u h } 0 À }σ σ h } 0 À }σ} m, 1 m k. In case of the Raviart Thomas elements, the approximation order is same to that of }u u h } 0. However, in case of BDM elements, }P h u u h } 0 gives an approximation which is one order higher than }u u h } 0 with a sufficiently high regularity of solution. If the domain Ω is nice, in the sense that the elliptic regularity property holds, then there is another technique to obtain superconvergence of }P h u u h } 0 for the RTN element Σ h. Consider the solution w of (18) Suppose that w P h u u h, in Ω, w 0, on BΩ. }w} 2 À }P h u u h } 0,

7 POISSON EQUATIONS 7 by assuming that Ω has the elliptic regularity property. Unfortunately, it is known that this is not the case if Ω is a nonconvex polygon but the elliptic regularity is true when BΩ is sufficiently regular or Ω is a convex polygonal domain. Now we can use the duality argument (or the Aubin Nitsche trick) to achieve superconvergence of }P h u u h } 0. To do it, let ξ w and consider (19) (20) pξ, τq pdiv τ, wq 0, τ P Σ, pdiv ξ, vq pp h u u h, vq, v P V. If we take τ σ σ h, v u u h, and add them, we have pξ, σ σ h q pdivpσ σ h q, wq pdiv ξ, u u h q By the Galerkin orthogonality in (15 16), we have pp h u u h, u u h q }P h u u h } 2 0. pξ Π h ξ, σ σ h q pdivpσ σ h q, w P h wq pdivpξ Π h ξq, u u h q }P h u u h } 2 0. Observe that divpξ Π h ξq 0 because div ξ P h u u h P h div ξ div Π h ξ. Furthermore, div σ f and div σ h P h div σ P h f. Hence (21) }P h u u h } 2 0 pξ Π h ξ, σ σ h q pf P h f, w P h wq. The rate of superconvergence depends on the regularity of f, k, and Σ h. If k 0 and f P H 1 pωq, then }ξ Π h ξ} 0 À h}ξ} 1 À h}p h u u h } 0, }w P h w} 0 À h}w} 1 À h}p h u u h } 0, }f P h f} 0 À h}f} 1. Combining these estimates with (21), we have }P h u u h } 0 Oph 2 q. If k 1, f P H 1 pωq and Σ h is the BDM element, then }w P h w} 0 À h 2 }w} 2 À h 2 }P h u u h } 0. Using this and some estimates we used in k 0 case, we can obtain }P h u u h } 0 Oph 3 q from (21). As a special case, we may consider k 0, Σ h BDM 1, and f P h f 0. In this case, (21) gives }P h u u h } 0 Oph 3 q, which is a supersuperconvergence result Post-processing of u h. Based on the superconvergence of }P h u u h } 0, one can use a local post-processing to achieve a numerical solution of u with higher accuracy. Here we use the Stenberg post-processing. For V h P k pt h q let V h P k 2pT h q, and define a post-processed numerical solution u h by (22) (23) pu h, 1q T pu h, 1q T, p u h, vq T pσ h, vq T, v P P k 2 pt q,

8 8 JEONGHUN LEE for all T P T h. It is not difficult to check that u h is well-defined by this definition. Note that this post-processing is a local procedure, which has a very small computational cost. We now prove that }u u h } 0 has an accuracy as good as that of }P h u u h } 0. Theorem 3. Suppose that }σ σ h } 0 À h k l }σ} k l, and }P h u u h } 0 Oph k l 1 q with l 0 or l 1. For the u h defined by (22) and (23), }u u h } 0 À h k l 1. Proof. Let Vh 1 P 0pT h q, V h P k l 1pT h q and define Ṽh be the orthogonal complement of Vh 1 in V h. Define P h 1, P h, P h by the orthogonal L 2 projections of L 2 pωq into Vh 1, V h, Ṽh, respectively. By the triangle inequality and a result }u P h u} 0 À h k l 1 }u} k l 1 of the Bramble Hilber lemma, it is enough to estimate }u h P h u} 0. Decomposing u h into u h u1 h ũ h P Vh 1 ` Ṽh, we have (24) u h P h u pu1 h P 1 h uq pũ h P h uq P V 1 h ` Ṽh. Since }u 1 h P 1 h u} 0 }u h P h u} 0 Oph k l 1 q, we only need to take care of }ũ h P h u} 0 by the triangle inequality. Now we recall that u σ and on each triangle T, p u, h wq pσ, h wq, w P Ṽh, in which h is the element-wise gradient. Considering the difference of this equation and (23) on Ω, and using (24), we obtain p h pp h u ũ h q, hwq p h pu u h q, hwq pσ σ h, h wq, w P V h, p h pp h u u h q, h wq p h pp h u uq, hwq pσ σ h, h wq, w P V h. By taking w P h u u h, one can obtain (25) } h pp h u u h q} 0 } h pp h u u h q} 0 } h pp h u uq} 0 }σ σ h } 0 À h 1 }P h u u h } 0 h k l }u} k l 1 À h 1 }P h u u h } 0 h k l }u} k l 1, h k l }σ} k l where the second inequality is obtained by the inverse inequality, the Bramble Hilbert lemma, the assumption on }σ σ h } 0, and the last inequality is due to σ u. Note that h pp h u u h q hp P h u ũ h q because h pp 1 h u u1 h q 0. Furthermore, } P h u ũ h } 0 À h} h p P h u ũ h q} 0, by a variant of the Friedrichs inequality, or a scaling argument. Combining this with (25), we have } P h u ũ h } 0 À }P h u u h } 0 h k l 1 }u} k l 1 À h k l 1.

9 POISSON EQUATIONS 9 The assumptions of the above theorem with l 1, hold when V h P k pt h q and Σ h is the corresponding stable BDM element. In fact, }σ σ h } 0 À h k 2 }σ} k 2 holds in this case. However, one can see that the convergence order of }u u h } 0 is limited by }P h u u h } 0 in the above proof, so this higher order accuracy of σ error does not improve the accuracy of u error with u h. The assumptions with l 0 are typically obtained when Σ h is the RTN element, and elliptic regularity for the duality argument is available Inhomogeneous boundary conditions. In this section we consider mixed Poisson problem with inhomogeneous mixed boundary conditions. Let Γ D, Γ N be open subsets of BΩ such that Γ D YΓ N BΩ, Γ D YΓ N H. Let us consider a problem (26) u f in Ω, Bu (27) u g D on Γ D, Bn g N on Γ N. We recall the variational form of this problem in the primal method. Let V D tv P H 1 pωq v ΓD g D u, V 0 tv P H 1 pωq v ΓD 0u. Note that, for the solution u of (26 27) and for all v P V 0, Bu pf, vq p u, vq p u, vq v ds p u, vq g N v ds, BΩ Bn Γ N because Bu{Bn g N on Γ N and v ΓD 0. Inspired from this observation, we define V h,d V D Y P k pt h q, V h,0 V 0 Y P k pt h q, and the variational form is to seek u h P V h,d such that p u h, vq pf, vq g N v ds : pf, vq xg N, vy ΓN, v P V h,0. Γ N Here the Dirichlet boundary condition g D is essentially imposed on the function space V h,d whereas the Neumann boundary condition g N is naturally satisfied by being a solution of the above equation. Thus the Dirichlet boundary condition is called an essential boundary condition and the Neumann boundary condition is called a natural boundary condition for this problem. Now we discuss mixed Poisson problem with these boundary conditions. We first observe that the Neumann boundary condition is equivalent to σ n g N. Define Σ N tτ P Hpdiv, Ωq τ n ΓN g N u, Σ 0 tτ P Hpdiv, Ωq τ n ΓN 0u.

10 10 JEONGHUN LEE For σ u with the solution of (26 27) u, and for all τ P Σ 0, we have pσ, τq p u, τq pu, div τq uτ n ds pu, div τq g D pτ nq ds. BΩ Γ D Defining Σ h,n Σ h Y Σ N, Σ h,0 Σ h Y Σ 0, the variational form is to seek pσ h, u h q P Σ h,n V h such that pσ h, τq pu h, div τq g D pτ nq ds, τ P Σ h,0, Γ D pdiv σ h, vq pf, vq, v P V h. In contrast to the primal method, the Neumann boundary condition is an essential boundary condition and the Dirichlet boundary condition is a natural boundary condition in mixed Poisson problems. 4. Implementation with FEniCS In this section we show how to implement mixed finite element methods for the Poisson equation with FEniCS, an open source software for automated finite element computation. For introduction of FEniCS we recommend the FEniCS tutorial written by H. P. Langtangen. ( tutorial/index.html) A PDF format of the tutorial and all source codes of examples are also available in the linked site. In the following code mixed finite element methods of the Poisson problem with the homogeneous Dirichlet boundary condition is implemented with Python interface of FEniCS. """ Mixed Poisson problem implementation (homogeneous Dirichlet B/C). Jeonghun J. Lee Last modified """ # Import dolfin and numpy modules from dolfin import * import numpy # Exact solution u_ex = Expression( x[0]*(1-x[0])*sin(pi*x[1]) ) sigma_ex = Expression(( (1-2*x[0])*sin(pi*x[1]), \ pi*x[0]*(1-x[0])*cos(pi*x[1]) )) f = Expression( -sin(pi*x[1])*(2+pi*pi*x[0]*(1-x[0])) ) # L2 norm of error computation def errnorm(u_exact, u, W):

11 POISSON EQUATIONS 11 u_w = interpolate(u, W) u_ex_w = interpolate(u_exact, W) e_u = Function(W) e_u.vector()[:] = u_ex_w.vector().array() - u_w.vector().array() error = e_u**2*dx return assemble(error, mesh=w.mesh()) # Degree of finite elements for velocity deg = 1 # Memories for element sizes and errors h = [] # element sizes E = [] # errors # Solve problems with refining mesh sizes for nx in [2,4,8,16,32]: # Create mesh and define function spaces mesh = UnitSquareMesh(nx,nx) Mh = FunctionSpace(mesh, "RT", deg+1) Vh = FunctionSpace(mesh, "DG",deg) V = MixedFunctionSpace([Mh, Vh]) # Build linear system (tau, v) = TestFunctions(V) (sigma, u) = TrialFunctions(V) a = inner(sigma, tau)*dx + u*div(tau)*dx + div(sigma)*v*dx L = f*v*dx # Solve linear system su = Function(V) solve(a == L, su) (sigma,u) = su.split() # Compute L2 errors W1 = FunctionSpace(mesh, "CG", deg+3) W2 = VectorFunctionSpace(mesh, "CG", deg+3) u_error = sqrt(errnorm(u_ex, u, W1)) sigma_error = sqrt(errnorm(sigma_ex, sigma, W2)) # Store errors and mesh data errors = \{ u : u_error, sigma : sigma_error\} h.append(nx) E.append(errors) # list of dicts

12 12 JEONGHUN LEE # Compute convergence rates from math import log as ln # log exists in dolfin module error_types = E[0].keys() for error_type in sorted(error_types): l = len(e) print \backslash nl2 norm of, error_type, error for i in range(0, l): if i==0: print Mesh size= 1/%g Error= %.3e \ Convergence rate= -- % (h[i], E[i][error_type]) else: r = ln(e[i][error_type]/e[i-1][error_type])/ln(0.5) print Mesh size= 1/%g Error= %.3e \ Convergence rate= %.2f % (h[i], E[i][error_type], r) Here is a code for a problem with inhomogeneous boundary conditions. In this problem domain is the unit square and exact solutions are u e x e y xy 1, σ pe x e y y, e x e y xq, with f 2e x e y. A mixed boundary condition is given by the Dirichlet boundary condition on x 0 part of the boundary and the Neumann boundary condition on the rest of the boundary. """ Mixed Poisson problem implementation (inhomogeneous mixed B/C). Jeonghun J. Lee Last modified """ # Import dolfin and numpy modules from dolfin import * import numpy # Exact solution u_0 = Expression( exp(x[1]) + 1 ) u_ex = Expression( exp(x[0])*exp(x[1]) + x[0]*x[1] + 1 ) sigma_ex = Expression(( exp(x[0])*exp(x[1]) + x[1], \ exp(x[0])*exp(x[1]) + x[0] )) f = Expression( 2*exp(x[0])*exp(x[1]) ) # L2 norm of error computation def errnorm(u_exact, u, W): u_w = interpolate(u, W) u_ex_w = interpolate(u_exact, W) e_u = Function(W) e_u.vector()[:] = u_ex_w.vector().array() - u_w.vector().array() error = e_u**2*dx return assemble(error, mesh=w.mesh())

13 POISSON EQUATIONS 13 # Degree of finite elements for Vh deg = 1 # Storage for element sizes and errors h = [] # element sizes E = [] # errors tol = 1E-14 # error tolerance # Solve problems with refining mesh sizes for nx in [2,4,8,16,32]: # Create mesh and define function spaces mesh = UnitSquareMesh(nx,nx) Mh = FunctionSpace(mesh, "BDM", deg+1) Vh = FunctionSpace(mesh, "DG",deg) V = MixedFunctionSpace([Mh, Vh]) # Boundary condition n = FacetNormal(mesh) class Nbdy(SubDomain): def inside(self, x, on_boundary): return on_boundary and ((abs(x[1]-1.) < tol) \ or (abs(x[0]-1.) < tol) or (abs(x[1]) < tol)) bc = DirichletBC(V.sub(0), sigma_ex, Nbdy()) # Build a linear system (tau, v) = TestFunctions(V) (sigma, u) = TrialFunctions(V) a = inner(sigma, tau)*dx + u*div(tau)*dx + div(sigma)*v*dx L = f*v*dx + u_0*inner(tau, n)*ds # Solve the linear system su = Function(V) solve(a == L, su, bc) (sigma, u) = su.split() # Compute L2 errors W1 = FunctionSpace(mesh, "CG", deg+3) W2 = VectorFunctionSpace(mesh, "CG", deg+3) u_error = sqrt(errnorm(u_ex, u, W1)) up_error = sqrt(errnorm(up, u, W1)) sigma_error = sqrt(errnorm(sigma_ex, sigma, W2)) # Store errors and mesh data

14 14 JEONGHUN LEE errors = { u : u_error, Phu - uh : up_error, sigma : sigma_error} h.append(nx) E.append(errors) # list of dicts # Compute convergence rates from math import log as ln # log exists in dolfin module error_types = E[0].keys() for error_type in sorted(error_types): l = len(e) print \backslash nl2 norm of, error_type, error for i in range(0, l): if i==0: print Mesh size= 1/%g Error= %.3e \ Convergence rate= -- % (h[i], E[i][error_type]) else: r = ln(e[i][error_type]/e[i-1][error_type])/ln(0.5) print Mesh size= 1/%g Error= %.3e \ Convergence rate= %.2f % (h[i], E[i][error_type], r) (To be appended) Last updated : 2013, October 1.