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## Minimal Surface Problem on Unit Disk

This example shows how to solve a nonlinear elliptic problem.

### A Nonlinear PDE

A nonlinear problem is one whose coefficients not only depend on spatial coordinates, but also on the solution itself. An example of this is the minimal surface equation

` `

on the unit disk, with on the boundary. To express this equation in toolbox form, note that the elliptic equation in toolbox syntax is

` `

The PDE coefficient `c` is the multiplier of , namely

` ` is a function of the solution , so the problem is nonlinear. In toolbox syntax, you see that the and coefficients are 0.

### Geometry

Create a PDE Model with a single dependent variable, and include the geometry of the unit disk. The `circleg` function represents this geometry. Plot the geometry and display the edge labels.

```numberOfPDE = 1; model = createpde(numberOfPDE); geometryFromEdges(model,@circleg); pdegplot(model,'EdgeLabels','on'); axis equal title 'Geometry with Edge Labels';``` ### Specify PDE 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);```

### Boundary Conditions

Create a function that represents the boundary condition .

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

### Generate Mesh

```generateMesh(model,'Hmax',0.1); figure; pdemesh(model); axis equal``` ### Solve PDE

Because the problem is nonlinear, solvepde invokes nonlinear solver. Observe the solver progress by setting the `SolverOptions.ReportStatistics` property of `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 Solution

```figure; pdeplot(model,'XYData',u,'ZData',u); xlabel 'x' ylabel 'y' zlabel 'u(x,y)' title 'Minimal surface'``` ##### Support Get trial now