poisolv

(Not recommended) Fast solver for Poisson's equation on rectangular grid

poisolv is not recommended. To solve Poisson's equations, use solvepde. For details, see Solve Problems Using PDEModel Objects.

Syntax

u = poisolv(model,p,e,t,f)

u = poisolv(b,p,e,t,f)

Description

u = poisolv(model,p,e,t,f) solves a Poisson's equation $$\Delta u=f$$ on a regular rectangular [p,e,t] mesh. The model must have only Dirichlet boundary conditions. A combination of sine transforms and tridiagonal solutions is used for increased performance.

u = poisolv(b,p,e,t,f) solves a Poisson's equation with Dirichlet boundary conditions u = b on a regular rectangular [p,e,t] mesh.

Examples

Fast Poisson Solver

Solve the Poisson's equation $-\Delta \mathit{u}=3{\mathit{x}}^2$ on a square domain with Dirichlet boundary conditions using the poisolv function.

Create a model object and include the square geometry created using the squareg function.

model = createpde;
g = @squareg;
geometryFromEdges(model,g);

Plot the geometry with the edge labels.

pdegplot(model,"EdgeLabels","on")
axis([-1.1 1.1 -1.1 1.1])

Apply the following Dirichlet boundary conditions. The solution is $0.2\cos \left(\pi \mathit{y}/2\right)$ on the right boundary (edge 2) and zero on all other boundaries.

innerBC = @(region,state) 0.2*cos(pi/2*region.y);
applyBoundaryCondition(model,"Dirichlet","Edge",2,"u",innerBC);
applyBoundaryCondition(model,"Dirichlet","Edge",[1 3 4],"u",0);

The fast Poisson solver requires a regular rectangular grid. Use the poimesh function to generate a mesh meeting this requirement. Plot the mesh.

[p,e,t] = poimesh(g,16);
figure; 
pdemesh(p,e,t); 
axis equal

Specify the PDE coefficients.

c = 1;
a = 0;
f = '3*x.^2';

Solve the equation on different meshes using the poisolv function.

for n = [16 32 64 128 256 512]
    [p,e,t] = poimesh(g,n);
    tic;
      u = poisolv(model,p,e,t,f);
    tfast = toc;
    fprintf('%-5d|%15.5g\n',n,tfast);
end

16   |         0.3334
32   |       0.059799
64   |       0.045932
128  |       0.045877
256  |        0.21496
512  |        0.38376

Plot the solution on the finest mesh.

figure;
pdeplot(p,[],t,"XYData",u,"ZData",u)

Input Arguments

model — PDE model

PDEModel object

PDE model, specified as a PDEModel object.

Example: model = createpde

b — Dirichlet boundary conditions for all boundary points

boundary matrix | boundary file

Dirichlet boundary conditions for all boundary points, specified as a boundary matrix or boundary file. Pass a boundary file as a function handle or as a file name. A boundary matrix is generally an export from the PDE Modeler app.

Example: b = 'circleb1', b = "circleb1", or b = @circleb1

Data Types: double | char | string | function_handle

p — Mesh points

matrix

Mesh points, specified as a 2-by-Np matrix of points, where Np is the number of points in the mesh. For details on the mesh data representation, see initmesh.

Data Types: double

e — Mesh edges

matrix

Mesh edges, specified as a 7-by-Ne matrix of edges, where Ne is the number of edges in the mesh. For details on the mesh data representation, see initmesh.

Data Types: double

t — Mesh triangles

matrix

Mesh triangles, specified as a 4-by-Nt matrix of triangles, where Nt is the number of triangles in the mesh. For details on the mesh data representation, see initmesh.

Data Types: double

f — Right side of Poisson's equation

scalar | matrix | character vector | character array | string scalar | string vector | coefficient function

Right side of a Poisson's equation, specified as a scalar, matrix, character vector, character array, string scalar, string vector, or coefficient function.

Data Types: double | char | string | function_handle

Output Arguments

u — PDE solution

vector

PDE solution, returned as a vector.

References

[1] Strang, G. Introduction to Applied Mathematics. Wellesley-Cambridge Press, Cambridge, MA, 1986, pp. 453–458.

Version History

Introduced before R2006a

R2016a: Not recommended

poisolv is not recommended. To solve Poisson's equations, use solvepde. For details, see Solve Problems Using PDEModel Objects. There are no plans to remove poisolv.