Poisson's Equation with Point Source and Adaptive Mesh Refinement

This example shows how to solve a Poisson's equation with a delta-function point source on the unit disk using the adaptmesh function.

Specifically, solve the Poisson's equation

$$-\Delta u=\delta (x,y)$$

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

$$u(r,\theta )=\frac{\log (r)}{2\pi },$$

which is singular at the origin.

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

The following variables define the 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 a geometry and include it in the model.

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

Plot the geometry and display the edge labels.

figure; 
pdegplot(model,"EdgeLabels","on"); 
axis equal
title("Geometry With Edge Labels Displayed")

Specify the zero solution at all four outer edges of the circle.

applyBoundaryCondition(model,"dirichlet","Edge",(1:4),"u",0);

adaptmesh solves elliptic PDEs using the adaptive mesh generation. The tripick parameter lets you 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 the finest mesh.

figure; 
pdemesh(p,e,t); 
axis equal

Plot the error values.

x = p(1,:)';
y = p(2,:)';
r = sqrt(x.^2+y.^2);
uu = -log(r)/2/pi;
figure;
pdeplot(p,e,t,"XYData",u-uu,"ZData",u-uu,"Mesh","off");

Plot the FEM solution on the finest mesh.

figure;
pdeplot(p,e,t,"XYData",u,"ZData",u,"Mesh","off");