Minimal Surface Problem

This example shows how to solve the minimal surface equation

on the unit disk $\Omega =\left\{\left(\mathit{x},\mathit{y}\right)|{\mathit{x}}^2+{\mathit{y}}^2\le 1\right\}$, with $$u(x,y)=x^2$$ on the boundary $\partial \Omega$. An elliptic equation in the toolbox form is

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

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 $$u(x,y)=x^2$$.

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          1.2144e-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'