Partial Differential Equation Toolbox |

A Minimal Surface Problem

In many problems the coefficients c, a, and f do not only depend on x and y, but also on the solution u itself. Consider the equation

on the unit disk
with on .

This problem is nonlinear and cannot be solved with the regular elliptic solver. Instead, the nonlinear solver `pdenonlin` is used.

Let us solve this minimal surface problem using the `pdetool` GUI and command-line functions.

Using the Graphical User Interface

Make sure that the application mode in the `pdetool` GUI is set to Generic Scalar. The problem domain is simply a unit circle. Draw it and move to the Boundary mode to define the boundary conditions. Use Select All from the Edit menu to select all boundaries. Then double-click a boundary to open the Boundary Condition dialog box. The Dirichlet condition u = x2 is entered by typing `x.^2` into the r edit box. Next, open the PDE Specification dialog box to define the PDE. This is an elliptic equation with

The nonlinear c is entered into the c edit box as

-     1./sqrt(1+ux.^2+uy.^2)

Initialize a mesh and refine it once.

Before solving the PDE, select Parameters . . . from the Solve menu and check the Use nonlinear solver option. Also, set the tolerance parameter to `0.001`.

Click the = button to solve the PDE. Use the Plot Selection dialog box to plot the solution in 3-D (check `u` and `continuous` selections in the Height column) to visualize the saddle shape of the solution.

Using Command-Line Functions

Working from the command line, the following sequence of commands solves the minimal surface problem and plots the solution. The M-files `circleg` and `circleb2` contain the geometry specification and boundary condition functions, respectively.

-     g='circleg';
      b='circleb2';
      c='1./sqrt(1+ux.^2+uy.^2)';
      rtol=1e-3;

      [p,e,t]=initmesh(g);
      [p,e,t]=refinemesh(g,p,e,t);

      u=pdenonlin(b,p,e,t,c,0,0,'Tol',rtol);

      pdesurf(p,t,u)

You can run this example by typing `pdedemo3`.

| A Scattering Problem |  | Domain Decomposition |