This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English verison of the page.

Note: This page has been translated by MathWorks. Please click here
To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

Poisson's Equation with Point Source and Adaptive Mesh Refinement

This example solves a Poisson's equation with a delta-function point source on the unit disk using the adaptmesh function in the Partial Differential Equation Toolbox™.

Specifically, we solve the Poisson's equation

on the unit disk with zero Dirichlet boundary conditions. The exact solution expressed in polar coordinates is

which is singular at the origin.

By using adaptive mesh refinement, the Partial Equation Toolbox™ can accurately find the solution everywhere away from the origin.

Problem Definition

The following variables will define our problem:

  • c, a: The coefficients of the PDE.

  • f: A function that captures a point source at the origin. It returns 1/area for the triangle containing the origin and 0 for other triangles.

c = 1;
a = 0;
f = @circlef;

Create a PDE Model with a single dependent variable

numberOfPDE = 1;
model = createpde(numberOfPDE);

% Create geometry and append to the PDE Model
% For more information, please see the documentation page for |circleg| and |pdegeom|.
g = @circleg;

Boundary Conditions

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

axis equal
title 'Geometry With Edge Labels Displayed';

% Solution is zero at all four outer edges of the circle

Generate Mesh

adaptmesh solves elliptic PDEs using adaptive mesh generation. The 'tripick' parameter lets the user specify a function that returns which triangles will be refined in the next iteration. circlepick returns triangles with computed error estimates greater a given tolerance. The tolerance is provided to circlepick using the 'par' parameter.

[u,p,e,t] = adaptmesh(g,model,c,a,f,'tripick','circlepick','maxt',2000,'par',1e-3);
Number of triangles: 258
Number of triangles: 515
Number of triangles: 747
Number of triangles: 1003
Number of triangles: 1243
Number of triangles: 1481
Number of triangles: 1705
Number of triangles: 1943
Number of triangles: 2155

Maximum number of triangles obtained.

Plot Finest Mesh

axis equal

Plot Error

x = p(1,:)';
y = p(2,:)';
r = sqrt(x.^2+y.^2);
uu = -log(r)/2/pi;

Plot FEM Solution on Finest Mesh


Was this topic helpful?