Project: Linear Finite Element Methods

The purpose of this project is to implement the finite element method for solving the Poisson equation in a general polygonal domain using the piecewise linear finite element.

Step 1: Download and install iFEM

  • Clone or download the zip file of iFEM from GitHub.
  • If a zip folder is dowloaded, unzip the file to where you like.
  • In MATLAB, go to the iFEM folder .
  • Run setpath.m.

Step 2: Mesh

% Generate mesh for the unit square
[node,elem] = squaremesh([0,1,0,1],0.25);
showmesh(node,elem);

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% Generate mesh for the unit disk
[node,elem] = circlemesh(0,0,1,0.2);
showmesh(node,elem);

 - Min quality 0.7570 - Mean quality 0.9696 - Uniformity 4.34%

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% Uniformly refine it to get a finer mesh
[node,elem] = uniformrefine(node,elem);
showmesh(node,elem);

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Step 3: Assembling the matrix

Compare three ways of assembling the stiffness matrix discussed in Progamming of Finite Element Methods.

Record the time by

tic; assemblingstandard; toc;
tic; assemblingsparse; toc;
tic; assembling; toc;

Compare the computational time for different N by applying the uniform refinement of the initial mesh several times. Try to test the assembling until N > 5*1e^4.

Step 4: Right hand side

Using three points quadrature (i.e. 3 middle points of a triangle) to compute the right hand side vector.

Step 5: Boundary conditions

  • Use findboundary.m to get all boundary nodes and edges
  • Code pure Dirichlet boundary condition $u = g_D$
  • Code pure Neumann boundary condition $\nabla u\cdot n = g_N$
  • (optional) Code Robin boundary condition $u + d\nabla u\cdot n = g_R$

Step 6: Verify convergence

  • Choose a smooth solution, say $u = \sin(2\pi x)\cos(2\pi y)$, calculate the right hand side $f$ and boundary conditions for the unit square.

  • Use your subroutine to get an approximation and use showresult to plot the mesh and the solution.

  • Use uniformrefine to refine the mesh and compute a sequence of solutions.

  • Compute the error in $H^1$-norm and $L^2$-norm using getH1error and getL2error.

  • Use the stiffness matrix to compute the error $|\nabla(u_I - u_h)|$, where $u_I$ is the nodal interpolation.

  • Use showrateh to plot the rate of convergence of these error.

  • Test both Dirichlet and Neumann problems.

Step 7: A challenging problem

Code your subroutine in a general way such that you can solve the Poisson equation on a different mesh by changing the input arguments.

After you get the desireable results for the unit square, try to solve $-\Delta u = 1$ with constant Neumann boundary conditions on the unit disk. The exact solution can be found using a separation of variable in the polar coordinate.

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