Poisson's Equation on Unit Disk

Open Live Script

This example shows how to numerically solve a Poisson's equation, compare the numerical solution with the exact solution, and refine the mesh until the solutions are close.

The Poisson equation on a unit disk with zero Dirichlet boundary condition can be written as $- Δ u = 1$ in $Ω$ , $u = 0$ on $δ Ω$ , where $Ω$ is the unit disk. The exact solution is

$u (x, y) = \frac{1 - x^{2} - y^{2}}{4} .$

For most PDEs, the exact solution is not known. However, the Poisson's equation on a unit disk has a known, exact solution that you can use to see how the error decreases as you refine the mesh.

Problem Definition

Create the PDE model and include the geometry.

model = createpde(); geometryFromEdges(model,@circleg);

Plot the geometry and display the edge labels for use in the boundary condition definition.

figure pdegplot(model,'EdgeLabels','on'); axisequal

Figure contains an axes object. The axes object contains 5 objects of type line, text.

Specify zero Dirichlet boundary conditions on all edges.

applyBoundaryCondition(model,'dirichlet',...'Edge',1:model.Geometry.NumEdges,...'u',0);

Specify the coefficients.

specifyCoefficients(model,'m',0,'d',0,'c',1,'a',0,'f',1);

Solution and Error with a Coarse Mesh

Create a mesh with target maximum element size 0.1.

hmax = 0.1; generateMesh(model,'Hmax',hmax); figure pdemesh(model); axisequal

Figure contains an axes object. The axes object contains 2 objects of type line.

Solve the PDE and plot the solution.

results = solvepde(model); u = results.NodalSolution; pdeplot(model,'XYData',u) title('Numerical Solution'); xlabel('x') ylabel('y')

Figure contains an axes object. The axes object with title Numerical Solution contains an object of type patch.

Compare this result with the exact analytical solution and plot the error.

p = model.Mesh.Nodes; exact = (1 - p(1,:).^2 - p(2,:).^2)/4; pdeplot(model,'XYData'u -精确的)title('Error'); xlabel('x') ylabel('y')

Figure contains an axes object. The axes object with title Error contains an object of type patch.

Solutions and Errors with Refined Meshes

Solve the equation while refining the mesh in each iteration and comparing the result with the exact solution. Each refinement halves theHmaxvalue. Refine the mesh until the infinity norm of the error vector is less than ${5 \cdot 10}^{- 7}$ .

hmax = 0.1; error = []; err = 1;whileerr > 5e-7% run until error <= 5e-7generateMesh(model,'Hmax',hmax);% refine meshresults = solvepde(model); u = results.NodalSolution; p = model.Mesh.Nodes; exact = (1 - p(1,:).^2 - p(2,:).^2)/4; err = norm(u - exact',inf);% compare with exact solutionerror = [error err];% keep history of errhmax = hmax/2;end

Plot the infinity norm of the error vector for each iteration. The value of the error decreases in each iteration.

plot(error,'rx','MarkerSize',12); ax = gca; ax.XTick = 1:numel(error); title('Error History'); xlabel('Iteration'); ylabel('Norm of Error');

Figure contains an axes object. The axes object with title Error History contains an object of type line.

Plot the final mesh and its corresponding solution.

figure pdemesh(model); axisequal

Figure contains an axes object. The axes object contains 2 objects of type line.

figure pdeplot(model,'XYData',u) title('Numerical Solution'); xlabel('x') ylabel('y')

Figure contains an axes object. The axes object with title Numerical Solution contains an object of type patch.

Compare the result with the exact analytical solution and plot the error.

p = model.Mesh.Nodes; exact = (1 - p(1,:).^2 - p(2,:).^2)/4; pdeplot(model,'XYData'u -精确的)title('Error'); xlabel('x') ylabel('y')

Figure contains an axes object. The axes object with title Error contains an object of type patch.