Minimal Surface Problem

This example shows how to solve the minimal surface equation

-(11+|u|2u)=0

on the unit disk Ω={(x,y)|x2+y21}, with u(x,y)=x2 on the boundary Ω.

This example uses the programmatic workflow. For the PDE Modeler app solution, see Minimal Surface Problem: PDE Modeler App.

An elliptic equation in the toolbox form is

-(cu)+au=f.

Therefore, for the minimal surface problem, the coefficients are as follows:

c=11+|u|2,a=0,f=0.

Because the coefficient c is a function of the solution u, the minimal surface problem is a nonlinear elliptic problem.

To solve the minimal surface problem using the programmatic workflow, first create a PDE model with a single dependent variable.

numberOfPDE = 1;
model = createpde(numberOfPDE);

Create the geometry and include it in the model. The circleg function represents this geometry.

geometryFromEdges(model,@circleg);

Plot the geometry displaying the edge labels.

pdegplot(model,'EdgeLabels','on'); 
axis equal
title 'Geometry with Edge Labels';

Specify the coefficients.

a = 0;
f = 0;
cCoef = @(region,state) 1./sqrt(1+state.ux.^2 + state.uy.^2);
specifyCoefficients(model,'m',0,'d',0,'c',cCoef,'a',a,'f',f);

Specify the boundary conditions using the function u(x,y)=x2.

bcMatrix = @(region,~)region.x.^2;
applyBoundaryCondition(model,'dirichlet',...
                             'Edge',1:model.Geometry.NumEdges,...
                             'u',bcMatrix);

Generate a mesh.

generateMesh(model,'Hmax',0.1);
figure; 
pdemesh(model); 
axis equal

Solve the problem by using the solvepde function. Because the problem is nonlinear, solvepde invokes a nonlinear solver. Observe the solver progress by setting the SolverOptions.ReportStatistics property of the model to 'on'.

model.SolverOptions.ReportStatistics = 'on';
result = solvepde(model);
Iteration     Residual     Step size  Jacobian: Full
   0          1.8540e-02
   1          2.8715e-04   1.0000000
   2          1.2145e-06   1.0000000
u = result.NodalSolution;

Plot the solution.

figure; 
pdeplot(model,'XYData',u,'ZData',u);
xlabel 'x'
ylabel 'y'
zlabel 'u(x,y)'
title 'Minimal Surface'