Poisson Equation Finite Element Method

Poisson Equation Finite Element Method

This tutorial will introduce you to the functionality for solving a PDE. Other introductions can be found by checking out DiffEqTutorials.jl. This tutorial assumes you have read the Ordinary Differential Equations tutorial.

In this example we will solve the Poisson Equation $Δu=f$. For our example, we will take the linear equation where $f(x,y) = \sin(2πx)\cos(2πy)$. For this equation we know that the solution is $u(x,y,t)= \sin(2πx)\cos(2πy)/(8π^2)$ with gradient $Du(x,y) = [\cos(2πx)\cos(2πy)/(4π) -\sin(2πx)\sin(2πy)/(4π)]$. Thus, we define the functions for a PoissonProblem as follows:

f(x) = sin(2π.*x[:,1]).*cos(2π.*x[:,2])
gD(x) = sin(2π.*x[:,1]).*cos(2π.*x[:,2])/(8π*π)

Or we can use the @fem_def macro to beautify our code. The first argument is the function signature, which here is (x). Second it's a list of variables to convert. This makes more sense in the Heat Equation examples, so we put in the blank expression () for now. Then we put in our expression, and lastly we define the parameter values. @fem_def will automatically replace x by x[:,1] and y by x[:,2], and will also substitute in the defined parameters. The previous definition using @fem_def is as follows:

f  = @fem_def((x),TestF,begin
  sin(α.*x).*cos(α.*y)
end,α=>6.28)
gD = @fem_def (x) TestgD begin
  sin(α.*x).*cos(α.*y)/β
end α=>6.28) β=>79.0

The linebreaks are not required but I think it makes it more legible!

Here we chose the Dirichlet boundary condition gD to give the theoretical solution. Other example problems can be found in src/examples/exampleProblems.jl. To solve this problem, we first have to generate a mesh. Here we will simply generate a mesh of triangles on the square [0,1]x[0,1] with dx=2^(-5). To do so, we use the code:

dx = 1//2^(5)
mesh = notime_squaremesh([0 1 0 1],dx,:dirichlet)
prob = PoissonProblem(f,mesh,gD=gD)

Note that by specifying :dirichlet, our boundary conditions is set on all boundaries to Dirichlet. This gives an FEMmesh object which stores a finite element mesh in the same layout as iFEM. Notice this code shows that the package supports the use of rationals in meshes. Other numbers such as floating point and integers can be used as well. Finally, to solve the equation we use

sol = solve(prob)

solve takes in a mesh and a PoissonProblem and uses the solver to compute the solution. Here the solver was chosen to be GMRES. Other solvers can be found in the documentation. This returns a FEMSolution object which holds data about the solution, such as the solution values (u). To plot the solution, we use the command

using Plots
plot(sol)

Here is the plot shown against the analytical solution to show the accuracy:

Poisson Example Solution