% femcode.m
% [p,t,b] from distmesh tool
% make sure your matlab path includes the directory where distmesh is installed.
fd=@(p) sqrt(sum(p.^2,2))-1;
[p,t]=distmesh2d(fd,@huniform,0.20,[-1,-1;1,1],[]);
be=boundedges(p,t);
b=unique(be);
f=vectorize(inline('0','x','y'));
k=vectorize(inline('1','x','y'));
u=vectorize(inline('3*y*x^2-y^3','x','y'));
ux=vectorize(inline('6*y*x','x','y'));
uy=vectorize(inline('3*x^2-3*y^2','x','y'));
% [K,F] = assemble(p,t) % K and F for any mesh of triangles: linear phi's
N=size(p,1);T=size(t,1); % number of nodes, number of triangles
% p lists x,y coordinates of N nodes, t lists triangles by 3 node numbers
K=sparse(N,N); % zero matrix in sparse format: zeros(N) would be "dense"
F=zeros(N,1); % load vector F to hold integrals of phi's times load f(x,y)
for e=1:T % integration over one triangular element at a time
nodes=t(e,:); % row of t = node numbers of the 3 corners of triangle e
Pe=[ones(3,1),p(nodes,:)]; % 3 by 3 matrix with rows=[1 xcorner ycorner]
Area=abs(det(Pe))/2; % area of triangle e = half of parallelogram area
C=inv(Pe); % columns of C are coeffs in a+bx+cy to give phi=1,0,0 at nodes
% now compute 3 by 3 Ke and 3 by 1 Fe for element e
qpt = (1/3)*sum(p(nodes,:));
I = feval(k,qpt(1),qpt(2));
grad=C(2:3,:);Ke=Area*I*grad'*grad; % element matrix from slopes b,c in grad
g = feval(f,qpt(1),qpt(2));
Fe=Area/3*g; % integral of phi over triangle is volume of pyramid: f(x,y)=1
% multiply Fe by f at centroid for load f(x,y): one-point quadrature!
% centroid would be mean(p(nodes,:)) = average of 3 node coordinates
K(nodes,nodes)=K(nodes,nodes)+Ke; % add Ke to 9 entries of global K
F(nodes)=F(nodes)+Fe; % add Fe to 3 components of load vector F
end % all T element matrices and vectors now assembled into K and F
% [Kb,Fb] = dirichlet(K,F,b) % assembled K was singular! K*ones(N,1)=0
% Implement Dirichlet boundary conditions U(b)=0 at nodes in list b
K(b,:)=0; [theta,rho] = cart2pol(p(b,:)); F(b)=sin(theta.*3); % set rows in K and F for boundary nodes.
K(b,b)=speye(length(b),length(b)); % put I into boundary submatrix of K
Kb=K; Fb=F; % Stiffness matrix Kb (sparse format) and load vector Fb
% Solving for the vector U will produce U(b)=0 at boundary nodes
U=Kb\Fb; % The FEM approximation is U_1 phi_1 + ... + U_N phi_N
% Plot the FEM approximation U(x,y) with values U_1 to U_N at the nodes
trisurf(t,p(:,1),p(:,2),0*p(:,1),U,'edgecolor','k','facecolor','interp');
view(2),axis([-1 1 -1 1]),axis equal,colorbar
% Calculate Energy Norms
enorm = 0;
for e=1:T % integration over one triangular element at a time
nodes=t(e,:); % row of t = node numbers of the 3 corners of triangle e
Pe=[ones(3,1),p(nodes,:)]; % 3 by 3 matrix with rows=[1 xcorner ycorner]
Area=abs(det(Pe))/2; % area of triangle e = half of parallelogram area
% C=inv(Pe); % columns of C are coeffs in a+bx+cy to give phi=1,0,0 at nodes
% now compute 3 by 3 Ke and 3 by 1 Fe for element e
% perform 3 point quadrature over triangle
lambda = zeros(3,3);
lambda(1,1) = 2/3;
lambda(1,2) = 1/6;
lambda(1,3) = 1/6;
lambda(2,1) = 1/6;
lambda(2,2) = 2/3;
lambda(2,3) = 1/6;
lambda(3,1) = 1/6;
lambda(3,2) = 1/6;
lambda(3,3) = 2/3;
qpts = lambda*p(nodes(1,[1:3]),:); % three gaussian quadrature points inside triangle element
w = ones(3,1)/3;
kvals = feval(k,qpts(:,1),qpts(:,2));
uxvals = feval(ux,qpts(:,1),qpts(:,2));
uyvals = feval(uy,qpts(:,1),qpts(:,2));
enorm = enorm + Area*((w.*kvals)'*(uxvals.^2 + uyvals.^2));
end % all T element matrices and vectors now assembled into K and F
nrm = sqrt(enorm);
enorm = 0;
for e=1:T % integration over one triangular element at a time
nodes=t(e,:); % row of t = node numbers of the 3 corners of triangle e
Pe=[ones(3,1),p(nodes,:)]; % 3 by 3 matrix with rows=[1 xcorner ycorner]
Area=abs(det(Pe))/2; % area of triangle e = half of parallelogram area
av = Pe\U(nodes);
% perform 3 point quadrature over triangle
lambda = zeros(3,3);
lambda(1,1) = 2/3;
lambda(1,2) = 1/6;
lambda(1,3) = 1/6;
lambda(2,1) = 1/6;
lambda(2,2) = 2/3;
lambda(2,3) = 1/6;
lambda(3,1) = 1/6;
lambda(3,2) = 1/6;
lambda(3,3) = 2/3;
qpts = lambda*p(nodes(1,[1:3]),:); % three gaussian quadrature points inside triangle element
w = ones(3,1)/3;
kvals = feval(k,qpts(:,1),qpts(:,2));
uxvals = feval(ux,qpts(:,1),qpts(:,2));
uyvals = feval(uy,qpts(:,1),qpts(:,2));
enorm = enorm + Area*((w.*kvals)'*((uxvals - av(2)).^2 + (uyvals - av(3)).^2));
end % all T element matrices and vectors now assembled into K and F
err = sqrt(enorm);
disp(['Energy Norm: ',num2str(err)]);
disp(['Energy Norm of exact solution: ',num2str(nrm)]);
disp(['Relative energy norm error in computed solution: ',num2str(err/nrm)])
u2 = feval(u,p(:,1),p(:,2));
u3 = U - u2;
L2exact = norm(u2,2);
L2 = norm(u3,2);
disp(['L2 Norm: ',num2str(L2)]);
disp(['L2 Norm of exact solution: ',num2str(L2exact)]);
disp(['Relative L2 norm error in computed solution: ',num2str(L2/L2exact)])