Computational Methods for Engineering Applications II. Homework Problem Sheet 4

Transcription

1 S. Mishra K. O. Lye L. Scarabosio Autumn Term 2015 Computational Methods for Engineering Applications II ETH Zürich D-MATH Homework Problem Sheet 4 This assignment sheet contains some tasks marked as Core problems. If you hand them in (see deadline at the end of the problem sheet), these tasks will be corrected. Full mark for the total of the core problems in all assignments will give a 20% bonus on the total points in the final exam. This is really a bonus, which means that at the exam you can still get the highest grade without having the bonus points (of course then you need to score more points at the exam). The total number of points for the Core problems of this sheet is 8 points. (The total number of points over all assignments will be around 20). Problem 4.1 We consider the problem where f L 2 (). Linear Finite Elements for the Poisson equation in 2D u = f(x) in R 2 (4.1.1) u(x) = 0 on (4.1.2) In the folder unittest you can find routines to test your implementation tasks for this problem. (4.1a) Write the variational formulation for (4.1.1)-(4.1.2). Solution: We multiply both the left handside and right handside of (4.1.1) by a test function v. Applying Green s formula for integration by parts on the left handside we get: u u(x)v(x) dx = u(x) v(x) dx (x)v(x) dx. n Since u satisfies Dirichlet boundary conditions, the test functions belong to H 1 0() and thus the boundary integral in the above expression vanishes. The variational formulation results then: Find u V = { w H 1 () : w = g on } such that u(x) v(x) dx = f(x)v(x) dx for all v H0(), 1 where the function g represents the boundary data (in our case g 0). We solve (4.1.1)-(4.1.2) by means of Galerkin discretization based on piecewise linear finite elements on triangular meshes of. Let us denote by ϕ i N, i = 0,..., N 1 the finite element Problem Sheet 4 Page 1 Problem 4.1

2 basis functions (hat functions) associated to the vertices of a given mesh, with N = N V the total number of vertices. The finite element solution u N to (4.1.1) can thus be expressed as u N (x) = N 1 i=0 µ i ϕ i N(x), (4.1.3) where µ = {µ i } N 1 i=0 is the vector of coefficients. Notice that we don t know µ i if i is an interior vertex, but we know that µ i = 0 if i is a vertex on the boundary. HINT: Here and in the following, we use zero-based indices in contrast to the lecture notes. Inserting ϕ i N, i = 0,..., N 1 as test functions in the variational formulation from subproblem (4.1a) we obtain the linear system of equations with A R N N and F R N. (4.1b) Write an expression for the entries of A and F in (4.1.4). Solution: We have A ij = for i, j = 0,..., N 1. Aµ = F, (4.1.4) ϕ j N (x) ϕi N(x) dx and F i = f(x)ϕ i N(x) dx, (4.1c) (Core problem) Complete the template file shape.hpp implementing the function inline double lambda(int i, double x, double y) which computes the the value a local shape function λ i (x), with i that can assume the values 0, 1 or 2, on the reference element depicted in Fig. 4.1 at the point x = (x, y). The convention for the local numbering of the shape functions is that λ i (x j ) = δ i,j, i, j = 0, 1, 2, with δ i,j denoting the Kronecker delta. Solution: See Listing 4.1 for the code. 1 #pragma once Listing 4.1: Implementation for lambda 2 3 //! The shape function (on the reference element) 4 //! 5 //! We have three shape functions. 6 //! 7 //! lambda(0, x, y) should be 1 in the point (0,0) and zero in (1,0) and (0,1) 8 //! lambda(1, x, y) should be 1 in the point (1,0) and zero in (0,0) and (0,1) 9 //! lambda(2, x, y) should be 1 in the point (0,1) and zero in (0,0) and (1,0) 10 //! Problem Sheet 4 Page 2 Problem 4.1

3 y 1 2 ˆK x Figure 4.1: Reference element ˆK for 2D linear finite elements. 11 i integer between 0 and 2 (inclusive). Decides which shape function to return. 12 x x coordinate in the reference element. 13 y y coordinate in the reference element. 14 i n l i n e double lambda ( i n t i, double x, double y ) { 15 i f ( i == 0) { 16 return 1 x y ; 17 } e l s e i f ( i == 1) { 18 return x ; 19 } e l s e { 20 return y ; 21 } 22 } (4.1d) (Core problem) Complete the template file grad shape.hpp implementing the function inline Eigen::Vector2d gradientlambda(const int i, double x, double y) which returns the value of the derivatives (i.e. the gradient) of a local shape functions λ i (x), with i that can assume the values 0, 1 or 2, on the reference element depicted in Fig. 4.1 at the point x = (x, y). Solution: See Listing 4.2 for the code. 1 #pragma once 2 # i n c l u d e <Eigen / Core> 3 Listing 4.2: Implementation for gradientlambda 4 //! The gradient of the shape function (on the reference element) 5 //! 6 //! We have three shape functions 7 //! 8 i integer between 0 and 2 (inclusive). Decides which shape function to return. Problem Sheet 4 Page 3 Problem 4.1

4 9 x x coordinate in the reference element. 10 y y coordinate in the reference element. 11 i n l i n e Eigen : : Vector2d gradientlambda ( c o n s t i n t i, double x, double y ) { 12 return Eigen : : Vector2d( 1 + ( i > 0) + ( i ==1), ( i > 0) + ( i ==2) ) ; 14 } The routine makecoordinatetransform contained in the file coordinate transform.hpp computes the Jacobian matrix of the linear map Φ l : R 2 R 2 such that ( ) ( ) ( ) ( ) 1 a11 0 a21 Φ l = = a 0 1, Φ l = = a 1 2, a 12 where a 1, a 2 R 2 are the two input arguments. (Core problem) Complete the template file stiffness matrix. hpp implementing the rou- (4.1e) tine template<class MatrixType, class Point> void computestiffnessmatrix(matrixtype& stiffnessmatrix, const Point& a, const Point& b, const Point& c) that returns the element stiffness matrix for the bilinear form associated to (4.1.1) and for the triangle with vertices a, b and c. HINT: Use the routine gradientlambda from subproblem (4.1d) to compute the gradients and the routine makecoordinatetransform to transform the gradients and to obtain the area of a triangle. Solution: See Listing 4.3 for the code. 1 #pragma once a 22 Listing 4.3: Implementation for computestiffnessmatrix 2 # i n c l u d e <Eigen / Core> 3 # i n c l u d e <Eigen / Dense> 4 # i n c l u d e g r a d s h a p e. hpp 5 # i n c l u d e c o o r d i n a t e t r a n s f o r m. hpp 6 # i n c l u d e i n t e g r a t e. hpp 7 8 //! Evaluate the stiffness matrix on the triangle spanned by 9 //! the points (a, b, c). 10 //! 11 //! Here, the stiffness matrix A is a 3x3 matrix 12 //! 13 //! A ij = λ K i (x, y) λ K j (x, y) dv 14 //! 15 //! where K is the triangle spanned by (a, b, c). 16 //! K Problem Sheet 4 Page 4 Problem 4.1

5 17 stiffnessmatrix should be a 3x3 matrix 18 //! At the end, will contain the integrals above. 19 //! 20 a the first corner of the triangle 21 b the second corner of the triangle 22 c the third corner of the triangle 23 template <c l a s s MatrixType, c l a s s Point > 24 void computestiffnessmatrix ( MatrixType& s t i f f n e s s M a t r i x, c o n s t Point& a, c o n s t Point& b, c o n s t Point& c ) 25 { 26 Eigen : : Matrix2d coordinatetransform = makecoordinatetransform ( b a, c a ) ; 27 double volumefactor = std : : abs ( coordinatetransform. determinant ( ) ) ; 28 Eigen : : Matrix2d elementmap = coordinatetransform. inverse ( ). transpose ( ) ; 29 f o r ( i n t i = 0; i < 3; ++ i ) { 30 f o r ( i n t j = i ; j < 3; ++ j ) { s t i f f n e s s M a t r i x ( i, j ) = i n t e g r a t e ( [ & ] ( double x, double y ) { 33 Eigen : : Vector2d gradlambdai = elementmap gradientlambda ( i, x, y ) ; 34 Eigen : : Vector2d gradlambdaj = elementmap gradientlambda ( j, x, y ) ; return volumefactor gradlambdai. dot ( gradlambdaj ) ; }) ; 39 } 40 } // Make symmetric (we did not need to compute these value above) 43 f o r ( i n t i = 0; i < 3; ++ i ) { 44 f o r ( i n t j = 0; j < i ; ++ j ) { 45 s t i f f n e s s M a t r i x ( i, j ) = s t i f f n e s s M a t r i x ( j, i ) ; 46 } 47 } } The routine integrate in the file integrate. hpp uses a quadrature rule to compute the approximate value of ˆK f(ˆx) dˆx, where f is a function, passed as input argument. (4.1f) (Core problem) Complete the template file load vector.hpp implementing the routine Problem Sheet 4 Page 5 Problem 4.1

6 template<class Vector, class Point> void computeloadvector(vector& loadvector, const Point& a, const Point& b, const Point& c, const std::function<double(double, double)>& f) that returns the element load vector for the linear form associated to (4.1.1), for the triangle with vertices a, b and c, and where f is a function handler to the right-hand side of (4.1.1). HINT: Use the routine lambda from subproblem (4.1c) to compute values of the shape functions on the reference element, and the routines makecoordinatetransform and integrate from the handout to map the points to the physical triangle and to compute the integrals. Solution: See Listing 4.4 for the code. 1 #pragma once 2 # i n c l u d e <Eigen / Core> 3 # i n c l u d e <Eigen / Dense> Listing 4.4: Implementation for computeloadvector 4 # i n c l u d e c o o r d i n a t e t r a n s f o r m. hpp 5 # i n c l u d e i n t e g r a t e. hpp 6 # i n c l u d e shape. hpp 7 # i n c l u d e <f u n c t i o n a l > 8 9 //! Evaluate the load vector on the triangle spanned by 10 //! the points (a, b, c). 11 //! 12 //! Here, the load vector is a vector (v i ) of 13 //! three components, where 14 //! 15 //! v i = λ K i (x, y)f(x, y) dv 16 //! 17 //! where K is the triangle spanned by (a, b, c). 18 //! 19 loadvector should be a vector of length //! At the end, will contain the integrals above. 21 //! 22 a the first corner of the triangle 23 b the second corner of the triangle 24 c the third corner of the triangle 25 f the function f (LHS). 26 template <c l a s s Vector, c l a s s Point > 27 void computeloadvector ( Vector& loadvector, 28 c o n s t Point& a, c o n s t Point& b, c o n s t Point& c, 29 c o n s t std : : f u n c t i o n <double ( double, double )>& f ) 30 { K Problem Sheet 4 Page 6 Problem 4.1

7 31 Eigen : : Matrix2d coordinatetransform = makecoordinatetransform ( b a, c a ) ; 32 double volumefactor = std : : abs ( coordinatetransform. determinant ( ) ) ; f o r ( i n t i = 0; i < 3; ++ i ) { 35 loadvector ( i ) = i n t e g r a t e ( [ & ] ( double x, double y ) { 36 Eigen : : Vector2d z = coordinatetransform Eigen : : Vector2d ( x, y ) + Eigen : : Vector2d ( a ( 0 ), a ( 1 ) ) ; 37 return f ( z ( 0 ), z ( 1 ) ) lambda ( i, x, y ) volumefactor ; 38 }) ; 39 } } (4.1g) (Core problem) Complete the template file stiffness matrix assembly.hpp implementing the routine template<class Matrix> void assemblestiffnessmatrix(matrix& A, const Eigen::MatrixXd& vertices, const Eigen::MatrixXi& triangles) to compute the finite element matrix A as in (4.1.4). The input argument vertices is a N V 3 matrix of which the i-th row contains the coordinates of the i-th mesh vertex, i = 0,..., N V 1, with N V the number of vertices. The input argument triangles is a N T 3 matrix where the i-th row contains the indices of the vertices of the i-th triangle, i = 0,..., N T 1, with N T the number of triangles in the mesh. HINT: Use the routine computestiffnessmatrix from subproblem (4.1e) to compute the local stiffness matrix associated to each element. HINT: Use the sparse format to store the matrix A. Solution: See Listing 4.5 for the code. 1 #pragma once Listing 4.5: Implementation for assemblestiffnessmatrix 2 # i n c l u d e <Eigen / Core> 3 # i n c l u d e <Eigen / Sparse> 4 # i n c l u d e <vector > 5 6 # i n c l u d e s t i f f n e s s m a t r i x. hpp 7 8 //! Sparse Matrix type. Makes using this type easier. 9 t y p e d e f Eigen : : SparseMatrix<double> SparseMatrix ; //! Used for filling the sparse matrix. 12 t y p e d e f Eigen : : T r i p l e t <double> T r i p l e t ; Problem Sheet 4 Page 7 Problem 4.1

8 13 14 //! Assemble the stiffness matrix 15 //! for the linear system 16 //! 17 A will at the end contain the Galerkin matrix 18 vertices a list of triangle vertices 19 triangles a list of triangles 20 template <c l a s s Matrix > 21 void assemblestiffnessmatrix ( Matrix& A, c o n s t Eigen : : MatrixXd& v e r t i c e s, 22 c o n s t Eigen : : M a t r i x X i& t r i a n g l e s ) 23 { c o n s t i n t numberofelements = t r i a n g l e s. rows ( ) ; 26 A. r e s i z e ( v e r t i c e s. rows ( ), v e r t i c e s. rows ( ) ) ; std : : vector <T r i p l e t > t r i p l e t s ; t r i p l e t s. reserve ( numberofelements 3 3) ; f o r ( i n t i = 0; i < numberofelements ; ++ i ) { 33 auto& indexset = t r i a n g l e s. row ( i ) ; c o n s t auto& a = v e r t i c e s. row ( indexset ( 0 ) ) ; 36 c o n s t auto& b = v e r t i c e s. row ( indexset ( 1 ) ) ; 37 c o n s t auto& c = v e r t i c e s. row ( indexset ( 2 ) ) ; Eigen : : Matrix3d s t i f f n e s s M a t r i x ; 40 computestiffnessmatrix ( s t i f f n e s s M a t r i x, a, b, c ) ; f o r ( i n t n = 0; n < 3; ++n ) { 43 f o r ( i n t m = 0; m < 3; ++m) { 44 auto t r i p l e t = T r i p l e t ( indexset ( n ), indexset (m), s t i f f n e s s M a t r i x ( n, m) ) ; 45 t r i p l e t s. push back ( t r i p l e t ) ; 46 } 47 } 48 } A. s e t F r o m T r i p l e t s ( t r i p l e t s. begin ( ), t r i p l e t s. end ( ) ) ; 51 } (4.1h) (Core problem) Complete the template file load vector assembly.hpp implementing the routine void assembleloadvector(eigen::vectorxd& F, const Eigen::MatrixXd& vertices, const Eigen::MatrixXi& triangles, const std::function<double(double, double)>& f) Problem Sheet 4 Page 8 Problem 4.1

9 to compute the right-hand side vector F as in (4.1.4). The input arguments vertices and triangles are as in subproblem (4.1g), and f is an in subproblem (4.1f). HINT: Proceed in a similar way as for assemblestiffnessmatrix and use the routine computeloadvector from subproblem (4.1f). Solution: See Listing 4.6 for the code. 1 #pragma once Listing 4.6: Implementation for assembleloadvector 2 # i n c l u d e <Eigen / Core> 3 # i n c l u d e l o a d v e c t o r. hpp 4 5 //! Assemble the load vector into the full right hand side 6 //! for the linear system 7 //! 8 F will at the end contain the RHS values for each vertex. 9 vertices a list of triangle vertices 10 triangles a list of triangles 11 f the RHS function f. 12 void assembleloadvector ( Eigen : : VectorXd& F, 13 c o n s t Eigen : : MatrixXd& v e r t i c e s, 14 c o n s t Eigen : : M a t r i x X i& t r i a n g l e s, 15 c o n s t std : : f u n c t i o n <double ( double, double )>& f ) 16 { 17 c o n s t i n t numberofelements = t r i a n g l e s. rows ( ) ; F. r e s i z e ( v e r t i c e s. rows ( ) ) ; 20 F. setzero ( ) ; 21 f o r ( i n t i = 0; i < numberofelements ; ++ i ) { 22 c o n s t auto& indexset = t r i a n g l e s. row ( i ) ; c o n s t auto& a = v e r t i c e s. row ( indexset ( 0 ) ) ; 25 c o n s t auto& b = v e r t i c e s. row ( indexset ( 1 ) ) ; 26 c o n s t auto& c = v e r t i c e s. row ( indexset ( 2 ) ) ; Eigen : : Vector3d elementvector ; 29 computeloadvector ( elementvector, a, b, c, f ) ; f o r ( i n t i = 0; i < 3; ++ i ) { 32 F ( indexset ( i ) ) += elementvector ( i ) ; 33 } 34 } 35 } The routine Problem Sheet 4 Page 9 Problem 4.1

10 void setdirichletboundary(eigen::vectorxd& u, Eigen::VectorXi& interiorvertexindices, const Eigen::MatrixXd& vertices, const Eigen::MatrixXi& triangles, const std::function<double(double, double)>& g) implemented in the file dirichlet boundary.hpp provided in the handout does the following: it gets in input the matrices vertices and triangles as defined in subproblem (4.1g) and the function handle g to the boundary data, i.e. to g such that u = g on (in our case g 0); it returns in the vector interiorvertexindices the indices of the interior vertices, that is of the vertices that are not on the boundary ; if x i is a vertex on the boundary, then it sets u(i )=g(x i ), that is, in our case, it sets to 0 the entries of the vector u corresponding to vertices on the boundary. (4.1i) (Core problem) Complete the template file fem solve.hpp with the implementation of the function int solvefiniteelement(vector& u, const Eigen::MatrixXd& vertices, const Eigen::MatrixXi& triangles, const std::function<double(double, double)>& f, const std::function<double(double, double)>& g). This function takes in input the matrices vertices, triangles as defined in the previous subproblems, the function handle f to the right-hand side f in (4.1.1) and the function handle g to the boundary data. For the moment, consider g to be the zero function. The output argument u has to contain, at the end of the function, the finite element solution u N to (4.1.1). HINT: Use the routines assemblestiffnessmatrix and assembleloadvector from subproblems (4.1g) and (4.1h), respectively, to obtain the matrix A and the vector F as in (4.1.4), and then use the provided routine setdirichletboundary to set the boundary values of u to zero and to select the free degrees of freedom. Solution: See Listing 4.7 for the code. 1 #pragma once Listing 4.7: Implementation for solvefiniteelement 2 # i n c l u d e <Eigen / Core> 3 # i n c l u d e <s t r i n g > 4 # i n c l u d e < i g l / readstl. h> 5 # i n c l u d e < i g l / s l i c e. h> 6 # i n c l u d e < i g l / s l i c e i n t o. h> 7 # i n c l u d e s t i f f n e s s m a t r i x a s s e m b l y. hpp 8 # i n c l u d e l o a d v e c t o r a s s e m b l y. hpp 9 # i n c l u d e d i r i c h l e t b o u n d a r y. hpp t y p e d e f Eigen : : VectorXd Vector ; //! Solve the FEM system. 14 //! 15 u will at the end contain the FEM solution. 16 vertices list of triangle vertices for the mesh Problem Sheet 4 Page 10 Problem 4.1

11 17 triangles list of triangles (described by indices) 18 f the RHS f (as in the exercise) 19 g the boundary value (as in the exercise) 20 //! return number of degrees of freedom (without the boundary dofs) 21 i n t solvefiniteelement ( Vector& u, 22 c o n s t Eigen : : MatrixXd& v e r t i c e s, 23 c o n s t Eigen : : M a t r i x X i& t r i a n g l e s, 24 c o n s t std : : f u n c t i o n <double ( double, double )>& f, 25 c o n s t std : : f u n c t i o n <double ( double, double )>& g ) 26 { 27 SparseMatrix A ; 28 assemblestiffnessmatrix (A, v e r t i c e s, t r i a n g l e s ) ; Vector F ; 31 assembleloadvector ( F, v e r t i c e s, t r i a n g l e s, f ) ; u. r e s i z e ( v e r t i c e s. rows ( ) ) ; 34 u. setzero ( ) ; 35 Eigen : : VectorXi i n t e r i o r V e r t e x I n d i c e s ; 36 s e t D i r i c h l e t B o u n d a r y ( u, i n t e r i o r V e r t e x I n d i c e s, v e r t i c e s, t r i a n g l e s, g ) ; F = A u ; SparseMatrix A I n t e r i o r ; i g l : : s l i c e (A, i n t e r i o r V e r t e x I n d i c e s, i n t e r i o r V e r t e x I n d i c e s, A I n t e r i o r ) ; 43 Eigen : : SimplicialLDLT <SparseMatrix> s o l v e r ; Vector F I n t e r i o r ; i g l : : s l i c e ( F, i n t e r i o r V e r t e x I n d i c e s, F I n t e r i o r ) ; s o l v e r. compute ( A I n t e r i o r ) ; i f ( s o l v e r. i n f o ( )!= Eigen : : Success ) { 52 throw std : : r u n t i m e e r r o r ( Could n o t decompose t h e m a t r i x ) ; 53 } Vector u I n t e r i o r = s o l v e r. solve ( F I n t e r i o r ) ; i g l : : s l i c e i n t o ( u I n t e r i o r, i n t e r i o r V e r t e x I n d i c e s, u ) ; return i n t e r i o r V e r t e x I n d i c e s. size ( ) ; 60 Problem Sheet 4 Page 11 Problem 4.1

12 Figure 4.2: Solution plot for subproblem (4.1j). 61 } (4.1j) (Core problem) Run the routine solvesquare contained in the file square.hpp to compute the finite element solution to (4.1.1) when = [0, 1] 2 is the unit square, the forcing term is given by f(x) = 2π 2 sin(πx) sin(πy) and the mesh is square 5.mesh. Use then the routine plot on mesh.py to produce a plot of the solution. Solution: See Fig. 4.2 for the plot. The functions computel2difference and computeh1difference, contained in the files L2 norm.hpp and H1 norm.hpp, respectively, are defined as double computel2difference(const Eigen::MatrixXd& vertices, const Eigen::MatrixXi& triangles, const Eigen::VectorXd& u1, const std::function<double(double, double)>& u2) double computeh1difference(const Eigen::MatrixXd& vertices, const Eigen::MatrixXi& triangles, const Eigen::VectorXd& u1, const std::function<eigen::vector2d(double, double)>& u2grad). In both cases, the argument u1 is considered to be a vector corresponding to a linear finite element function u 1, and thus containing the value of this function in the vertices of a mesh. The argument u2 in computel2difference is a function handle to a scalar function u 2 which is known analytically. The argument u2grad in computeh1difference is a function handle to a function gradient u 2, supposed to be known analytically. Then the routines computel2difference and computeh1difference compute an approximation to u 1 u 2 L 2 () and u 1 u 2 H 1 () = u 1 u 2 L 2 (), respectively. The routine convergenceanalysis contained in the file convergence.hpp performs a convergence study calling the routines computel2difference and computeh1difference. This function writes a file with the number of degrees of freedom used at each convergence step, a file with the computed L 2 -error norms and a file with the computed H 1 -error seminorms. (4.1k) Run the routine convergenceanalysis to perform a convergence study for the finite element solution to (4.1.1)-(4.1.2). For both errors, make a double logarithmic plot of error versus number of degrees of freedom. Which convergence orders with respect to the number of degrees of freedom do you observe? Solution: See Fig. 4.3 for the plots. Problem Sheet 4 Page 12 Problem 4.1

13 u FEM u L u FEM u H Number of free vertices Number of free vertices Figure 4.3: Convergence plots for subproblem (4.1k). Doing a linear fitting of the logarithm of the number of degrees of freedom versus the logarithm of the errors, we observe a rate of convergence of around 1 for the L 2 -norm error, and a rate of around 0.5 with respect to the H 1 -seminorm of the error. These rates are with respect to the number of degrees of freedom. They correspond, respectively to rates of around 2 and 1 with respect to the mesh width, as expected from the theory. We now consider the problem u = f(x) in R 2 (4.1.5) u(x) = g(x) on (4.1.6) where f L 2 () and now g is a continuous function which is in general not zero on the boundary. (4.1l) Modify your implementation of the routine solvefiniteelement implemented in subproblem (4.1i) in order to compute the finite element solution to (4.1.5)-(4.1.6). HINT: Use the information contained in the output of the routine setdirichletboundary and use the offset function technique. Solution: See Listing 4.7, line 38. (4.1m) Run the routine solvel contained in the file Lshape.hpp to compute the finite element solution to (4.1.5)-(4.1.6) when is the L-shaped domain = ( 1, 1) 2 \((0, 1) ( 1, 0)), as depicted in Fig The forcing term is given by f(x) = 0, the boundary condition by g = u, with u the exact solution, which, in polar coordinates, is given by u(r, ϑ) = r 2 3 sin( 2 3 ϑ), for r 0 and ϑ [0, 3 2π]. The mesh used is Lshape 5.mesh. Use then the routine plot on mesh.py to produce a plot of the solution. Solution: See Fig. 4.5 for the plot. (4.1n) Run the routine convergenceanalysis to perform a convergence study for the finite element solution to (4.1.5)-(4.1.6). For both errors, make a double logarithmic plot of error versus number of degrees of freedom. Which convergence orders with respect to the number of degrees of freedom do you observe? Solution: See Fig. 4.6 for the plots. In this case we have a rate of convergence with respect to the number of degrees of freedom of around 0.66 for the L 2 -norm and 0.32 for the H 1 -seminorm of the error. We can observe that the convergence rates are lower than the ones in subproblem (4.1k). The reason is that the exact solution presents a singularity at Problem Sheet 4 Page 13 Problem 4.1

14 Figure 4.4: Domain for subproblems (4.1m)-(4.1n) Figure 4.5: Solution plot for subproblem (4.1m). Problem Sheet 4 Page 14 Problem 4.1

15 u FEM u L u FEM u H Number of free vertices Number of free vertices Figure 4.6: Convergence plots for subproblem (4.1n). the origin. The convergence orders that we have seen in subproblem (4.1k) are achieved only if the exact solution is smooth enough, which is not the case here. Problem 4.2 Green s Formula To derive the variational formulation from a PDE, we needed multi-dimensional integration by parts as expressed through Green s first formula. In this problem we study the derivation of Green s formula, thus practicing elementary vector analysis and the application of Gauss theorem. Now prove Green s formula for R 2 j v dx = where j C 1 () 2 and v C 1 (). HINT: Use Gauss theorem. where F C 1 () 2. F dx = Solution: The vectorized product rule takes the form which, when integrated, is j n v ds + j v dx, F n ds, (jv) = j v + v j, (jv) dx = The first integral fits Gauss theorem, so we get j n v ds = which is what we wanted to show. Published on November 25. To be submitted on December 11. Last modified on January 7, 2016 j v dx + v j dx. j v dx + v j dx Problem Sheet 4 Page 15 Problem 4.2