High Level API Finite Element Methods based on ExtendableGrids.jl (for grid management) and ExtendableFEMBase.jl (for finite element basis functions and dof management). It offers a ProblemDescription interface, that basically involves assigning Unknowns and Operators. Such operators usually stem from a weak formulation of the problem and mainly consist of three types that can be customized via kernel functions:
- BilinearOperator,
- LinearOperator,
- NonlinearOperator (that automatically assemble Newton's method by automatic differentiation)
The following minimal example demonstrates how to setup a Poisson problem.
using ExtendableFEM
using ExtendableGrids
# build/load any grid (here: a uniform-refined 2D unit square into triangles)
xgrid = uniform_refine(grid_unitsquare(Triangle2D), 4)
# create empty PDE description
PD = ProblemDescription()
# create and assign unknown
u = Unknown("u"; name = "potential")
assign_unknown!(PD, u)
# assign Laplace operator
assign_operator!(PD, BilinearOperator([grad(u)]; factor = 1e-3))
# assign right-hand side data
function f!(fval, qpinfo)
x = qpinfo.x # global coordinates of quadrature point
fval[1] = x[1] * x[2]
end
assign_operator!(PD, LinearOperator(f!, [id(u)]))
# assing boundary data (here: u = 0)
assign_operator!(PD, HomogeneousBoundaryData(u; regions = 1:4))
# discretise = choose FESpace
FEType = H1Pk{1,2,3} # cubic H1-conforming element with 1 component in 2D
FES = FESpace{FEType}(xgrid)
# solve
sol = solve!(Problem, [FES])
# plot
using PyPlot
plot(id(u), sol; Plotter = PyPlot)