# Boundary Layer (Elliptic)

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:

- One should always use anisotropically refined meshes for problems with boundary layers.
- hp-FEM is vastly superior to h-FEM with linear and quadratic elements.
- One should use not only spatially anisotropic elements, but also polynomial anisotropy (different polynomial orders in each direction) for problems in boundary layers.

## Model problem

Equation solved: Poisson equation

(1)

Domain of interest: Square .

Boundary conditions: zero Dirichlet.

## Exact solution

where is the exact solution of the 1D singularly-perturbed problem

in with zero Dirichlet boundary conditions. This solution has the form

## Right-hand side

Calculated by inserting the exact solution into the equation.

## Sample solution

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.

## Convergence comparison for isotropic refinements

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:

## Convergence comparison for anisotropic refinements

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:

## h-FEM (p=1): comparison of isotropic and anisotropic refinements

DOF convergence graphs:

CPU convergence graphs:

## h-FEM (p=2): comparison of isotropic and anisotropic refinements

DOF convergence graphs:

CPU convergence graphs:

## hp-FEM: comparison of isotropic and anisotropic refinements

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.