On a Discontinuous Galerkin Method for Surface PDEs

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1 On a Discontinuous Galerkin Method for Surface PDEs Pravin Madhavan (joint work with Andreas Dedner and Bjo rn Stinner) Mathematics and Statistics Centre for Doctoral Training University of Warwick Applied PDEs Seminar University of Warwick, Coventry, 22nd January 2013

boundaries, cell biology: phase separation on biomembranes, diffusion

2 Motivation - PDEs on Surfaces PDEs on surfaces arise in various areas, for instance materials science: enhanced species diffusion along grain boundaries, cell biology: phase separation on biomembranes, diffusion processes on plasma membranes, fluid dynamics: surface active agents.

3 Motivation - Numerical Approaches Numerical methods are based on: parametrisation of the surface (eg graph),

4 Motivation - Numerical Approaches Numerical methods are based on: parametrisation of the surface (eg graph), level set representation,

5 Motivation - Numerical Approaches Numerical methods are based on: parametrisation of the surface (eg graph), level set representation, diffuse interface representation (phase field methods),

6 Motivation - Numerical Approaches Numerical methods are based on: parametrisation of the surface (eg graph), level set representation, diffuse interface representation (phase field methods), this talk: intrinsic approach, approximation of the surface by triangulated surfaces

7 Motivation - Numerical Approaches Numerical methods are based on: parametrisation of the surface (eg graph), level set representation, diffuse interface representation (phase field methods), this talk: intrinsic approach, approximation of the surface by triangulated surfaces FEM methods: [ Dziuk 1988 ]: elliptic problem, [ Dziuk, Elliott 2007 ] (2 papers): parabolic problem (stationary/evolving surface). advection dominated problem / hp-adaptive refinement DG methods.

Motivation - Numerical Approaches Numerical methods are based on: parametrisation of the surface (eg graph), level set representation, diffuse interface representation (phase field methods), this

8 Motivation - Numerical Approaches Numerical methods are based on: parametrisation of the surface (eg graph), level set representation, diffuse interface representation (phase field methods), this talk: intrinsic approach, approximation of the surface by triangulated surfaces FEM methods: [ Dziuk 1988 ]: elliptic problem, [ Dziuk, Elliott 2007 ] (2 papers): parabolic problem (stationary/evolving surface). advection dominated problem / hp-adaptive refinement DG methods. FEM DG

9 Motivation - Numerical Approaches Numerical methods are based on: parametrisation of the surface (eg graph), level set representation, diffuse interface representation (phase field methods), this talk: intrinsic approach, approximation of the surface by triangulated surfaces FEM methods: [ Dziuk 1988 ]: elliptic problem, [ Dziuk, Elliott 2007 ] (2 papers): parabolic problem (stationary/evolving surface). advection dominated problem / hp-adaptive refinement DG methods.

10 Outline 1. Notation and Setting 2. DG Approximation 3. Convergence Proof 4. Numerical Tests

11 Outline 1. Notation and Setting 2. DG Approximation 3. Convergence Proof 4. Numerical Tests

12 Some Notation Hypersurface: Γ R 3 compact, smooth, simply connected, oriented, no boundary. Signed distance function: d : U R with U a thin tube around Γ. Unit normal: ν(ξ) = d(ξ), ξ Γ. Projection of R 3 onto the tangent space T ξ Γ, ξ Γ: P(ξ) := I ν(ξ) ν(ξ), ξ Γ. Surface gradient: For any function η : U R, Γ η := η η νν = P η = (D 1 η, D 2 η, D 3 η).

13 Strong Problem Formulation Laplace-Beltrami operator: 3 Γ η := Γ ( Γ η) = D i D i η. i=1 Strong problem: For a given function f : Γ R, find u : Γ R such that Γ u + u = f in Γ.

14 Strong Problem Formulation Laplace-Beltrami operator: 3 Γ η := Γ ( Γ η) = D i D i η. i=1 Strong problem: For a given function f : Γ R, find u : Γ R such that Γ u + u = f in Γ. For weak formulation: Integration by parts formula on surfaces: ( η Γ v = v Γ η + ηv κ ) + ηv µ Γ Γ Γ where µ: outer co-normal of Γ on Γ, κ: mean curvature vector.

15 Weak Problem Formulation Sobolev spaces: H m (Γ) := {u L 2 (Γ) : α Γ u L2 (Γ) α m} with corresponding norm 1/2 u H m (Γ) := α m α Γ u 2 L 2 (Γ). Problem (P Γ ): Find u V := H 1 (Γ) such that Γ u Γ v + uv da = fv da, v V. Γ Γ

16 Weak Problem Formulation Sobolev spaces: H m (Γ) := {u L 2 (Γ) : α Γ u L2 (Γ) α m} with corresponding norm 1/2 u H m (Γ) := α m α Γ u 2 L 2 (Γ). Problem (P Γ ): Find u V := H 1 (Γ) such that Γ u Γ v + uv da = fv da, v V. Γ Γ Theorem (Aubin 1982) If f L 2 (Γ) then there is a unique weak solution u V to (P Γ ) which satisfies u H 2 (Γ) C f L 2 (Γ).

17 Outline 1. Notation and Setting 2. DG Approximation 3. Convergence Proof 4. Numerical Tests

The vertices sit on Γ Γ h is its linear interpolation.

18 Triangulated Surfaces Γ is approximated by a polyhedral surface Γ h composed of planar triangles K h. The vertices sit on Γ Γ h is its linear interpolation. T h : Associated regular, conforming triangulation i.e. Γ h = K h. K h T h

19 DG Setting DG space: V h := { v h L 2 (Γ h ) : v h Kh P 1 (K h ) K h T h }. This space allows for jumps across edges, to be penalised in the DG method.

20 DG Setting DG space: V h := { v h L 2 (Γ h ) : v h Kh P 1 (K h ) K h T h }. This space allows for jumps across edges, to be penalised in the DG method. Set of edges: E h. Unit conormals: n + h, n h to K + h, K h on e h E h. Trace values: v ± h := v h K ± h for v h V h.

21 DG Setting DG space: V h := { v h L 2 (Γ h ) : v h Kh P 1 (K h ) K h T h }. This space allows for jumps across edges, to be penalised in the DG method. Set of edges: E h. Unit conormals: n + h, n h to K + h, K h on e h E h. Trace values: v ± h := v h K ± h for v h V h. DG norm: u h 2 1,h := u h 2 H 1 (K h ), u h 2,h := K h T h u h 2 DG(Γ h ) := u h 2 1,h + u h 2,h. e h E h h 1 e h u + h u h 2 L 2 (e h ),

22 DG Problem Problem (P DG Γ h ): Find u h V h such that aγ DG h (u h, v h ) = f h v h da h v h V h Γ h where f h is related to f (later) and aγ DG h (u h, v h ) := Γh u h Γh v h + u h v h da h K h T K h h with β eh h 1 e h. (u + h u h ) 1 e h E e h h 2 ( Γ h v + h n+ h Γ h v h n h ) ds h (v + h v h ) 1 e h E e h h 2 ( Γ h u + h n+ h Γ h u h n h ) ds h + β eh (u + h u h )(v + h v h ) ds h e h e h E h

23 DG Problem Problem (P DG Γ h ): Find u h V h such that aγ DG h (u h, v h ) = f h v h da h v h V h Γ h where f h is related to f (later) and aγ DG h (u h, v h ) := Γh u h Γh v h + u h v h da h K h T K h h with β eh h 1 e h. (u + h u h ) 1 e h E e h h 2 ( Γ h v + h n+ h Γ h v h n h ) ds h (v + h v h ) 1 e h E e h h 2 ( Γ h u + h n+ h Γ h u h n h ) ds h + β eh (u + h u h )(v + h v h ) ds h e h e h E h Interior penalty method [ Arnold 1982 ]: If β eh = ω eh he 1 h and ω eh big enough then aγ DG is coercive, and there is a unique solution u h h V h to problem (P DG Γ ) with h u h DG(Γh ) C f h L 2 (Γ h ).

24 DG Problem: Remark [ Arnold 1982 ] (classical) IP method: aγ DG h (u h, v h ) := Γh u h Γh v h + u h v h da h K h T K h h (u + h u h ) 1 e h E e h h 2 ( Γ h v + h n+ h Γ h v h n h ) ds h (v + h v h ) 1 e h E e h h 2 ( Γ h u + h n+ h Γ h u h n h ) ds h + β eh (u + h u h )(v + h v h ) ds h e h e h E h [ Arnold et al ] (standard) IP method: aγ DG h (u h, v h ) := Γh u h Γh v h + u h v h da h K h T K h h (u + h n+ h + u h n h ) 1 e h E e h h 2 ( Γ h v + h + Γ h v h ) ds h (v + h n+ h v h n h ) 1 e h E e h h 2 ( Γ h u + h + Γ h u h ) ds h + β eh (u + h n+ h + u h n h ) (v + h n+ h + v h n h ) ds h e h e h E h

25 The Lift Goal: Compare u H 2 (Γ) solving (P Γ ) with u h V h solving (P DG Γ h ), but Γ h Γ. Lift: For η : Γ h R define η l (ξ) := η(x) where x = ξ + d(x)ν(ξ)

26 The Lift Goal: Compare u H 2 (Γ) solving (P Γ ) with u h V h solving (P DG Γ h ), but Γ h Γ. Lift: For η : Γ h R define η l (ξ) := η(x) where x = ξ + d(x)ν(ξ) One-to-one relation between Γ and Γ h, write x = x(ξ) or ξ = ξ(x). Right hand side: Define f h so that fh l = f on Γ.

27 Lifted Objects Lifted triangles: K l h = ξ(k h) Γ. Conforming triangulation Th l, Γ = Kh l. K l h T l h Lifted edges: e l h := ξ(e h) E l h.

28 Lifted Objects Lifted triangles: K l h = ξ(k h) Γ. Conforming triangulation Th l, Γ = Kh l. K l h T l h Lifted edges: e l h := ξ(e h) E l h. Lifted DG space: V l h := {v l h L2 (Γ) : v l h (ξ) = v h(x(ξ)) with some v h V h }, norm: v l h 2 DG(Γ) := K l h T l h v l h 2 H 1 (K l h ) + e l h El h h 1 e l h v l,+ h v l, h 2 L 2 (e l h )

29 DG Bilinear Form on Γ Consider a DG Γ (u, v) := K l h T l h e l h El h e l h El h + e l h El h K l h e l h e l h e l h Γ u Γ v + uv da (u + u ) 1 2 ( Γv + n + Γ v n ) ds (v + v ) 1 2 ( Γu + n + Γ u n ) ds β e l h (u + u )(v + v ) ds Unit conormals to K l+ h and K l h on eh l El h : n+ = n T ξ Γ. Penalty parameters β e l h := βe h δ eh.

30 Convergence Statement Theorem (Dedner, M., Stinner 2012) Let u H 2 (Γ) and u h V h denote the solutions to (P Γ ) and (P DG Γ ), respectively. h Denote by uh l V h l the lift of u h onto Γ. Then u u l h L 2 (Γ) + h u ul h DG(Γ) Ch 2 f L 2 (Γ).

31 Outline 1. Notation and Setting 2. DG Approximation 3. Convergence Proof 4. Numerical Tests

32 Outline of the Proof 1. Starting: u u l h DG(Γ) u φ l h DG(Γ) + φ l h ul h DG(Γ), φ l h = I l h u interpolate.

33 Outline of the Proof 1. Starting: u u l h DG(Γ) u φ l h DG(Γ) + φ l h ul h DG(Γ), φ l h = I l h u interpolate. 2. Interpolation estimate [ Dziuk 1988 ]: u φ l h DG(Γ) = u φ l h H 1 (Γ) Ch u H 2 (Γ) ( ) Ch f L 2 (Γ).

34 Outline of the Proof 1. Starting: u u l h DG(Γ) u φ l h DG(Γ) + φ l h ul h DG(Γ), φ l h = I l h u interpolate. 2. Interpolation estimate [ Dziuk 1988 ]: u φ l h DG(Γ) = u φ l h H 1 (Γ) Ch u H 2 (Γ) ( ) Ch f L 2 (Γ). 3. Using coercivity in V l h : C l s φ l h ul h 2 DG(Γ) adg Γ (φl h ul h, φl h ul h ) = aγ DG (φl h u, φl h ul h ) + adg Γ (u ul h, φl h ul h ).

35 Outline of the Proof 1. Starting: u u l h DG(Γ) u φ l h DG(Γ) + φ l h ul h DG(Γ), φ l h = I l h u interpolate. 2. Interpolation estimate [ Dziuk 1988 ]: u φ l h DG(Γ) = u φ l h H 1 (Γ) Ch u H 2 (Γ) ( ) Ch f L 2 (Γ). 3. Using coercivity in V l h : C l s φ l h ul h 2 DG(Γ) adg Γ (φl h ul h, φl h ul h ) 4. Using boundedness in H 2 (Γ) + V l h : = aγ DG (φl h u, φl h ul h ) + adg Γ (u ul h, φl h ul h ). a DG Γ (φl h u, φl h ul h ) C l b( φ l h u DG(Γ) + h 2 u H 2 (Γ)) φ l h u l h DG(Γ).

36 Outline of the Proof 1. Starting: u u l h DG(Γ) u φ l h DG(Γ) + φ l h ul h DG(Γ), φ l h = I l h u interpolate. 2. Interpolation estimate [ Dziuk 1988 ]: u φ l h DG(Γ) = u φ l h H 1 (Γ) Ch u H 2 (Γ) ( ) Ch f L 2 (Γ). 3. Using coercivity in V l h : C l s φ l h ul h 2 DG(Γ) adg Γ (φl h ul h, φl h ul h ) 4. Using boundedness in H 2 (Γ) + V l h : = aγ DG (φl h u, φl h ul h ) + adg Γ (u ul h, φl h ul h ). a DG Γ (φl h u, φl h ul h ) C l b( φ l h u DG(Γ) + h 2 u H 2 (Γ)) φ l h u l h DG(Γ). 5. Estimating variational crime error: aγ DG (u ul h, φl h ul h ) Ch2 f L 2 (Γ) φl h ul h DG(Γ).

37 Outline of the Proof 1. Starting: u u l h DG(Γ) u φ l h DG(Γ) + φ l h ul h DG(Γ), φ l h = I l h u interpolate. 2. Interpolation estimate [ Dziuk 1988 ]: u φ l h DG(Γ) = u φ l h H 1 (Γ) Ch u H 2 (Γ) ( ) Ch f L 2 (Γ). 3. Using coercivity in V l h : C l s φ l h ul h 2 DG(Γ) adg Γ (φl h ul h, φl h ul h ) 4. Using boundedness in H 2 (Γ) + V l h : = aγ DG (φl h u, φl h ul h ) + adg Γ (u ul h, φl h ul h ). a DG Γ (φl h u, φl h ul h ) C l b( φ l h u DG(Γ) + h 2 u H 2 (Γ)) φ l h u l h DG(Γ). 5. Estimating variational crime error: 6. Concluding: aγ DG (u ul h, φl h ul h ) Ch2 f L 2 (Γ) φl h ul h DG(Γ). u u l h DG(Γ) (1 + C) φ l h u H 1 (Γ) + Ch2 u H 2 (Γ) + Ch2 f L 2 (Γ) Ch f L 2 (Γ).

38 Coercivity and Boundedness: Inverse Estimate 3. Using coercivity in V l h : C l s φl h ul h 2 DG(Γ) adg Γ (φl h ul h, φl h ul h ) 4. Using boundedness in H 2 (Γ) + V l h : = aγ DG (φl h u, φl h ul h ) + adg Γ (u ul h, φl h ul h ). a DG Γ (φl h u, φl h ul h ) C l b( φ l h u DG(Γ) + h 2 u H 2 (Γ)) φ l h u l h DG(Γ). Lemma (Inverse Estimate) Let w H 2 (Γ) and wh l V h l. Let K h l T h l. Then for sufficiently small h, ( ) 1 Γ (w + wh l ) 2 L 2 ( K h l ) C h Γ(w + wh l ) 2 L 2 (K h l ) + h w 2 H 2 (K h l ).

39 Coercivity and Boundedness: Inverse Estimate 3. Using coercivity in V l h : C l s φl h ul h 2 DG(Γ) adg Γ (φl h ul h, φl h ul h ) 4. Using boundedness in H 2 (Γ) + V l h : = aγ DG (φl h u, φl h ul h ) + adg Γ (u ul h, φl h ul h ). a DG Γ (φl h u, φl h ul h ) C l b( φ l h u DG(Γ) + h 2 u H 2 (Γ)) φ l h u l h DG(Γ). Lemma (Inverse Estimate) Let w H 2 (Γ) and wh l V h l. Let K h l T h l. Then for sufficiently small h, ( ) 1 Γ (w + wh l ) 2 L 2 ( K h l ) C h Γ(w + wh l ) 2 L 2 (K h l ) + h w 2 H 2 (K h l ). Proof. Trace theorem and a standard scaling argument on K h T h, lift estimate onto Kh l T h l using result in [ Demlow 2009 ] and apply geometric estimates.

40 Variational Crime Error: Geometric Estimates 5. Estimating variational crime error: aγ DG (u ul h, φl h ul h ) Ch2 f L 2 (Γ) φl h ul h DG(Γ). aγ DG (u ul h, w h l ) = ( ) ) 1 (R h P) Γ u l K h l T h l K h l h Γwh l + 1 u (1 lh δ w lh + 1δh fwh l da h + (u l+ e h l El e h l h ul h ) 1 ( Γ w l+ h (n + 1 P(I dh)n l+ ) h 2 δ eh h Γ w l h (n 1 P(I dh)n l ) ) h ds δ eh + (w l+ h w l h ) 1 ( Γ u l+ h 2 (n + 1 P(I dh)n l+ ) h δ eh e l h El h e l h Γ u l h (n 1 P(I dh)n l ) ) h ds. δ eh

41 Variational Crime Error: Geometric Estimates 5. Estimating variational crime error: aγ DG (u ul h, φl h ul h ) Ch2 f L 2 (Γ) φl h ul h DG(Γ). Lemma (Dziuk 1988 and Giesselman & Mueller 2012) d L (Γ) Ch 2, 1 δ h L (Γ) Ch 2, δ h : local area change, δ h da h = da, ν ν h L (Γ) Ch, ν, ν h : unit normals on Γ and Γ h, 1 δ eh L (E l h )) Ch2, δ eh : local length change, δ eh ds h = ds, n Pn l h L (E l h ) Ch2, P R h L (Γ) Ch 2 where R h := 1 δ h P(I dh)p h (I dh) with H = 2 d and P h = I ν h ν h.

42 Outline 1. Notation and Setting 2. DG Approximation 3. Convergence Proof 4. Numerical Tests

43 Software All simulations have been performed using the Distributed and Unified Numerics Environment (DUNE). Initial mesh generation made use of 3D surface mesh generation module of the Computational Geometry Algorithms Library (CGAL). Further information about DUNE and CGAL can be found respectively on and

44 Test Problem on Unit Sphere Surface Helmholtz equation: Γ u + u = f on the unit sphere Γ = {x R 3 : x = 1}. The right-hand side f is chosen such that u(x 1, x 2, x 3 ) = cos(2πx 1 ) cos(2πx 2 ) cos(2πx 3 ) is the exact solution.

45 EOCs for Sphere Test Problem Elements h L 2 -error L 2 -eoc DG-error DG-eoc

46 Visualisation, Sphere Test Problem Can easily work with non-conforming grid:

47 Test Problem on Dziuk Surface Solve the Helmholtz equation on the Dziuk surface Γ = {x R 3 : (x 1 x 2 3 )2 + x x2 3 = 1}. The right-hand side f is chosen such that is the exact solution. u(x) = x 1 x 2

48 EOCs for Dziuk Surface Elements h L 2 -error L 2 -eoc DG-error DG-eoc e

49 Test Problem on Enzensberger-Stern Surface Solve the Helmholtz equation on the Enzensberger-Stern surface Γ = {x R 3 : 400(x 2 y 2 + y 2 z 2 + x 2 z 2 ) (1 x 2 y 2 z 2 ) 3 40 = 0.} The right-hand side f is again chosen such that u(x) = x 1 x 2 is the exact solution.

50 EOCs for Enzensberger-Stern Surface Elements h L 2 -error L 2 -eoc DG-error DG-eoc e Tricky: The computation of the lifted points ξ(x) when refining the surface. The EOC rates thus are a bit more volatile.

51 Other Choices for the Conormals Generalise the bilinear form: ãγ DG h (u h, v h ) := (u + h u h ) 1 e h E e h h 2 ( Γ h v + h n+ e h Γh v h n e h ) ds h (v + h v h ) 1 e h 2 ( Γ h u + h n+ e h Γh u h n e h ) ds h +... e h E h Choice n e h n + e h Description 1 n h n h planar (non-sym) 2 n h n + h [ Arnold 1982 ] (sym pos-def) (n h n+ h ) 1 2 (n+ 1 h n h ) 2 (n h n+ h ) 2 1 averaged (sym pos-def) (n+ h n h ) 4 n + h n h [ Arnold et al ] (sym pos-def)

52 Comparison of Choices Ratio of L 2 and DG errors for the test problem on the Enzensberger-Stern surface, benchmark is the analysed choice 2.

53 Comparison of Choices Ratio of L 2 and DG errors for the test problem on the Enzensberger-Stern surface, benchmark is the analysed choice 2. The [ Arnold et al ] (standard) IP method did not converge!

54 Conclusion I The (classical) IP method from [ Arnold 1982 ] can be lifted to surfaces.

55 Conclusion I The (classical) IP method from [ Arnold 1982 ] can be lifted to surfaces. I A priori error analysis requires estimation of additional terms in comparison with [ Dziuk 1988 ].

56 Conclusion I The (classical) IP method from [ Arnold 1982 ] can be lifted to surfaces. I A priori error analysis requires estimation of additional terms in comparison with [ Dziuk 1988 ]. I The expected order of convergence is obtained and observed.

57 Conclusion I The (classical) IP method from [ Arnold 1982 ] can be lifted to surfaces. I A priori error analysis requires estimation of additional terms in comparison with [ Dziuk 1988 ]. I The expected order of convergence is obtained and observed. I Numerics suggest that the (standard) IP method from [ Arnold et al ] cannot be lifted to surfaces.

58 Conclusion I The (classical) IP method from [ Arnold 1982 ] can be lifted to surfaces. I A priori error analysis requires estimation of additional terms in comparison with [ Dziuk 1988 ]. I The expected order of convergence is obtained and observed. I Numerics suggest that the (standard) IP method from [ Arnold et al ] cannot be lifted to surfaces. Open: Analysis for nonconforming meshes, higher order methods (extend [ Demlow 2009 ]), hp-adaptive a posteriori error analysis and refinement (extend [ Demlow & Dziuk 2008 ] and [ Houston et al ]). I

59 Conclusion I The (classical) IP method from [ Arnold 1982 ] can be lifted to surfaces. I A priori error analysis requires estimation of additional terms in comparison with [ Dziuk 1988 ]. I The expected order of convergence is obtained and observed. I Numerics suggest that the (standard) IP method from [ Arnold et al ] cannot be lifted to surfaces. Open: Analysis for nonconforming meshes, higher order methods (extend [ Demlow 2009 ]), hp-adaptive a posteriori error analysis and refinement (extend [ Demlow & Dziuk 2008 ] and [ Houston et al ]). I ku uhl kdg (Γ) C X Kh Th 2 krh kl 2,L (wk ) ηk +k h h p βeh [uh ]k2l2 ( K 1/2 + higher order geometric terms. h)

Conclusion I The (classical) IP method from [ Arnold 1982 ] can be lifted to surfaces. I A priori error analysis requires estimation of additional terms in comparison with [ Dziuk 1988 ].

60 Conclusion I The (classical) IP method from [ Arnold 1982 ] can be lifted to surfaces. I A priori error analysis requires estimation of additional terms in comparison with [ Dziuk 1988 ]. I The expected order of convergence is obtained and observed. I Numerics suggest that the (standard) IP method from [ Arnold et al ] cannot be lifted to surfaces. Open: Analysis for nonconforming meshes, higher order methods (extend [ Demlow 2009 ]), hp-adaptive a posteriori error analysis and refinement (extend [ Demlow & Dziuk 2008 ] and [ Houston et al ]). I ku uhl kdg (Γ) C X Kh Th 2 krh kl 2,L (wk ) ηk +k h h p βeh [uh ]k2l2 ( K 1/2 + higher order geometric terms. Thanks for your attention! Acknowledgement: grant EP/H023364/1 h)

Scientific Computing WS 2017/2018. Lecture 18. Jürgen Fuhrmann Lecture 18 Slide 1

Scientific Computing WS 2017/2018 Lecture 18 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 18 Slide 1 Lecture 18 Slide 2 Weak formulation of homogeneous Dirichlet problem Search u H0 1 (Ω) (here,