# assempde

Assemble stiffness matrix and right side of PDE problem

## Syntax

```u = assempde(model,c,a,f)
u = assempde(model,c,a,f,u0)
u = assempde(model,c,a,f,u0,time)
u = assempde(model,c,a,f,time)

u = assempde(b,p,e,t,c,a,f)
u = assempde(b,p,e,t,c,a,f,u0)
u = assempde(b,p,e,t,c,a,f,u0,time)
u = assempde(b,p,e,t,c,a,f,time)

[K,F] = assempde(model,c,a,f)
[K,F] = assempde(model,c,a,f,u0)
[K,F] = assempde(model,c,a,f,u0,time)
[K,F] = assempde(model,c,a,f,time)

[K,F] = assempde(b,p,e,t,c,a,f)
[K,F] = assempde(b,p,e,t,c,a,f,u0)
[K,F] = assempde(b,p,e,t,c,a,f,u0,time)
[K,F] = assempde(b,p,e,t,c,a,f,u0,time,sdl)
[K,F] = assempde(b,p,e,t,c,a,f,time)
[K,F] = assempde(b,p,e,t,c,a,f,time,sdl)

[K,F,B,ud] = assempde(model,c,a,f)
[K,F,B,ud] = assempde(model,c,a,f,u0)
[K,F,B,ud] = assempde(model,c,a,f,u0,time)
[K,F,B,ud] = assempde(model,c,a,f,time)

[K,F,B,ud] = assempde(b,p,e,t,c,a,f)
[K,F,B,ud] = assempde(b,p,e,t,c,a,f,u0)
[K,F,B,ud] = assempde(b,p,e,t,c,a,f,u0,time)
[K,F,B,ud] = assempde(b,p,e,t,c,a,f,time)

[K,M,F,Q,G,H,R] = assempde(model,c,a,f)
[K,M,F,Q,G,H,R] = assempde(model,c,a,f,u0)
[K,M,F,Q,G,H,R] = assempde(model,c,a,f,u0,time)
[K,M,F,Q,G,H,R] = assempde(model,c,a,f,u0,time,sdl)
[K,M,F,Q,G,H,R] = assempde(model,c,a,f,time)
[K,M,F,Q,G,H,R] = assempde(model,c,a,f,time,sdl)

[K,M,F,Q,G,H,R] = assempde(b,p,e,t,c,a,f)
[K,M,F,Q,G,H,R] = assempde(b,p,e,t,c,a,f,u0)
[K,M,F,Q,G,H,R] = assempde(b,p,e,t,c,a,f,u0,time)
[K,M,F,Q,G,H,R] = assempde(b,p,e,t,c,a,f,u0,time,sdl)
[K,M,F,Q,G,H,R] = assempde(b,p,e,t,c,a,f,time)
[K,M,F,Q,G,H,R] = assempde(b,p,e,t,c,a,f,time,sdl)

u = assempde(K,M,F,Q,G,H,R)
[K1,F1] = assempde(K,M,F,Q,G,H,R)
[K1,F1,B,ud] = assempde(K,M,F,Q,G,H,R)
```

## Description

`assempde` is the basic Partial Differential Equation Toolbox™ function. It assembles a PDE problem by using the FEM formulation described in Elliptic Equations. The command `assempde` assembles the scalar PDE problem

$-\nabla \cdot \left(c\nabla u\right)+au=f,$

for (x,y) ∊ Ω, or the system PDE problem

$-\nabla \cdot \left(c\otimes \nabla u\right)+au=f.$

The command can optionally produce a solution to the PDE problem.

For the scalar case the solution vector u is represented as a column vector of solution values at the corresponding node points from `p`. For a system of dimension N with np node points, the first np values of `u` describe the first component of u, the following np values of `u` describe the second component of u, and so on. Thus, the components of u are placed in the vector `u` as N blocks of node point values.

`model` is a `PDEModel` object that incorporates the number of equations, geometry, mesh, and boundary conditions.

`u = assempde(b,p,e,t,c,a,f)` assembles and solves the PDE problem using the finite element method. The `[p,e,t]` arguments are the 2-D mesh data (see Mesh Data for [p,e,t] Triples: 2-D).

`b` describes the boundary conditions of the PDE problem. For the recommended way of specifying boundary conditions, see Specify Boundary Conditions Objects. For all methods of specifying boundary conditions, see Forms of Boundary Condition Specification.

`[K,F] = assempde(b,p,e,t,c,a,f)` assembles the PDE problem by approximating the Dirichlet boundary condition with stiff springs (see Systems of PDEs for details). `K` and `F` are the stiffness matrix and right-hand side, respectively. The solution to the FEM formulation of the PDE problem is `u = K\F`.

`[K,F,B,ud] = assempde(b,p,e,t,c,a,f)` assembles the PDE problem by eliminating the Dirichlet boundary conditions from the system of linear equations. `u1 = K\F` returns the solution on the non-Dirichlet points. The solution to the full PDE problem can be obtained as the MATLAB® expression ```u = B*u1+ud```.

`[K,M,F,Q,G,H,R] = assempde(b,p,e,t,c,a,f)` gives a split representation of the PDE problem.

`u = assempde(K,M,F,Q,G,H,R)` collapses the split representation into the single matrix/vector form, and then solves the PDE problem by eliminating the Dirichlet boundary conditions from the system of linear equations.

`[K1,F1] = assempde(K,M,F,Q,G,H,R)` collapses the split representation into the single matrix/vector form, by fixing the Dirichlet boundary condition with large spring constants.

`[K1,F1,B,ud] = assempde(K,M,F,Q,G,H,R)` collapses the split representation into the single matrix/vector form by eliminating the Dirichlet boundary conditions from the system of linear equations.

The optional list of subdomain labels, `sdl`, restricts the assembly process to the subdomains denoted by the labels in the list. You cannot include a `sdl` argument in a model with 3-D geometry. The optional input arguments `u0` and `time` are used for the nonlinear solver and time stepping algorithms, respectively. The tentative input solution vector `u0` has the same format as `u`.

## Examples

### 3-D Elliptic Problem

Solve a 3-D elliptic PDE using a PDE model.

Create a PDE model container, import a 3-D geometry description, and view the geometry.

```model = createpde; importGeometry(model,'Block.stl'); h = pdegplot(model,'FaceLabels','on'); h(1).FaceAlpha = 0.5;```

Set zero Dirichlet conditions on faces 1 through 4 (the edges). Set Neumann conditions with g = –1 on face 6 and g = 1 on face 5.

```applyBoundaryCondition(model,'Face',1:4,'u',0); applyBoundaryCondition(model,'Face',6,'g',-1); applyBoundaryCondition(model,'Face',5,'g',1);```

Set coefficients c = 1, a = 0, and f = 0.1.

```c = 1; a = 0; f = 0.1;```

Create a mesh and solve the problem.

```generateMesh(model); u = assempde(model,c,a,f);```

Plot the solution on the surface.

`pdeplot3D(model,'colormapdata',u);`

### L-Shaped Membrane

Solve the equation Δu = 1 on the geometry defined by the L-shaped membrane. Use Dirichlet boundary conditions u = 0 on ∂Ω. Finally plot the solution.

```[p,e,t] = initmesh('lshapeg','Hmax',0.2); [p,e,t] = refinemesh('lshapeg',p,e,t); u = assempde('lshapeb',p,e,t,1,0,1); pdesurf(p,t,u)```

### Poisson's Equation with Point Source

Consider Poisson's equation on the unit circle with unit point source at the origin. The exact solution

$u=-\frac{1}{2\pi }\mathrm{log}\left(r\right)$

is known for this problem. We define the function ```f = circlef(p,t,u,time)``` for computing the right-hand side. `circlef` returns zero for all triangles except for the one located at the origin; for that triangle it returns 1/a, where a is the triangle area. `pdedemo7` executes an adaptive solution for this problem.