Git reference: Benchmark layer-boundary.
This example is a singularly perturbed problem with known exact solution that exhibits a thin boundary layer The reader can use it to perform various experiments with adaptivity. The sample numerical results presented below imply that:
Equation solved: Poisson equation
(1)-\Delta u + K^2 u - f = 0.
Domain of interest: Square (-1, 1)^2.
Boundary conditions: zero Dirichlet.
u(x,y) = \hat u(x) \hat u(y)
where \hat u is the exact solution of the 1D singularly-perturbed problem
-u'' + K^2 u = K^2
in (-1,1) with zero Dirichlet boundary conditions. This solution has the form
\hat u (x) = 1 - [exp(Kx) + exp(-Kx)] / [exp(K) + exp(-K)];
Calculated by inserting the exact solution into the equation.
Below we present a series of convergence comparisons. Note that the error plotted is the true approximate error calculated wrt. the exact solution given above.
Let us first compare the performance of h-FEM (p=1), h-FEM (p=2) and hp-FEM with isotropic refinements:
Final mesh (h-FEM, p=1, isotropic refinements):
Final mesh (h-FEM, p=2, isotropic refinements):
Final mesh (hp-FEM, isotropic refinements):
DOF convergence graphs:
CPU convergence graphs:
Next we compare the performance of h-FEM (p=1), h-FEM (p=2) and hp-FEM with anisotropic refinements:
Final mesh (h-FEM, p=1, anisotropic refinements):
Final mesh (h-FEM, p=2, anisotropic refinements):
Final mesh (hp-FEM, anisotropic refinements):
DOF convergence graphs:
CPU convergence graphs:
DOF convergence graphs:
CPU convergence graphs:
DOF convergence graphs:
CPU convergence graphs:
In the hp-FEM one has two kinds of anisotropy – spatial and polynomial. In the following, “iso” means isotropy both in h and p, “aniso h” means anisotropy in h only, and “aniso hp” means anisotropy in both h and p.
DOF convergence graphs (hp-FEM):
CPU convergence graphs (hp-FEM):
The reader can see that enabling polynomially anisotropic refinements in the hp-FEM is equally important as allowing spatially anisotropic ones.