201 : Poisson-Problem

(source code)

This example computes the solution $u$ of the two-dimensional Poisson problem

\[\begin{aligned} -\Delta u & = f \quad \text{in } \Omega \end{aligned}\]

with right-hand side $f(x,y) \equiv xy$ and homogeneous Dirichlet boundary conditions on the unit square domain $\Omega$ on a given grid.

The computed solution for the default parameters looks like this:

module Example201_PoissonProblem

using ExtendableFEM
using ExtendableGrids

# define variables
u = Unknown("u"; name = "potential")

# define data functions
function f!(fval, qpinfo)
	fval[1] = qpinfo.x[1] * qpinfo.x[2]

function main(; μ = 1.0, nrefs = 4, order = 2, Plotter = nothing, kwargs...)
	# problem description
	PD = ProblemDescription()
	assign_unknown!(PD, u)
	assign_operator!(PD, BilinearOperator([grad(u)]; factor = μ, kwargs...))
	assign_operator!(PD, LinearOperator(f!, [id(u)]; kwargs...))
	assign_operator!(PD, HomogeneousBoundaryData(u; regions = 1:4))

	# discretize
	xgrid = uniform_refine(grid_unitsquare(Triangle2D), nrefs)
	FES = FESpace{H1Pk{1, 2, order}}(xgrid)

	# solve
	sol = solve(PD, FES; kwargs...)

	# plot
	plt = plot([id(u), grad(u)], sol; Plotter = Plotter)

	return sol, plt

end # module

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