Partial Differential Equation Toolbox
This example shows how to numerically solve a Poisson's equation using the assempde function in Partial Differential Equation Toolbox™.
The particular PDE is
on the unit disk with zero-Dirichlet boundary conditions. The exact solution is
For most partial differential equations, the exact solution is not known. In this example, however, we can use the known, exact solution to show how the error decreases as the mesh is refined.
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The following variables will define our problem:
g: A specification function that is used by initmesh. For more information, please see the documentation pages for circleg and pdegeom.
c, a, f: The coefficients and inhomogeneous term.
g = @circleg; c = 1; a = 0; f = 1;
Plot the geometry and display the edge labels for use in the boundary condition definition.
figure pdegplot(g, 'edgeLabels', 'on'); axis equal % Create a pde entity for a PDE with a single dependent variable numberOfPDE = 1; pb = pde(numberOfPDE); % Create a geometry entity pg = pdeGeometryFromEdges(g); % Solution is zero at all four outer edges of the circle pb.BoundaryConditions = pdeBoundaryConditions(pg.Edges(1:4), 'u', 0);
The function initmesh takes a geometry specification function and returns a discretization of that domain. The 'hmax' option lets the user specify the maximum edge length. In this case, because the domain is a unit disk, a maximum edge length of one creates a coarse discretization.
[p,e,t] = initmesh(g,'hmax',1); figure pdemesh(p,e,t); axis equal
We repeatedly refine the mesh until the infinity-norm of the error vector is less than a .
For this domain, each refinement halves the lengths of the edges of the triangles that compose the mesh. Note also that the error decreases by a factor of about one-third.
er = Inf; while er > 0.001 [p,e,t] = refinemesh(g,p,e,t); u = assempde(pb,p,e,t,c,a,f); exact = (1-p(1,:).^2-p(2,:).^2)'/4; er = norm(u-exact,'inf'); fprintf('Error: %e. Number of nodes: %d\n',er,size(p,2)); end
Error: 1.292265e-02. Number of nodes: 25 Error: 4.079923e-03. Number of nodes: 81 Error: 1.221020e-03. Number of nodes: 289 Error: 3.547924e-04. Number of nodes: 1089
figure pdemesh(p,e,t); axis equal
The scale of the vertical axis shows that the error is small and of the order .