# NIST-09 (Wave Front)¶

Commonly used for testing adaptive refinement algorithms is Poisson’s equation with a solution that has a steep wave front in the interior of the domain.

## Model problem¶

Equation solved: Poisson equation

(1)

Domain of interest: Unit Square

Boundary conditions: Dirichlet, given by exact solution.

## Exact solution¶

where , is the center of the circular wave front, is the distance from the wave front to the center of the circle, and gives the steepness of the wave front.

## Material parameters¶

This benchmark has four different versions, we use the global variable PARAM (below) to switch among them.

int PARAM = 3;     // PARAM determines which parameter values you wish to use
// for the steepness and location of the wave front.
//  | name       | ALPHA | X_LOC | Y_LOC | R_ZERO
// 0: mild         20      -0.05   -0.05    0.7
// 1: steep        1000    -0.05   -0.05    0.7
// 2: asymmetric   1000     1.5     0.25    0.92
// 3: well         50       0.5     0.5     0.25


## Sample solution¶

Solution for , , and :

## Comparison 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, h-anisotropic refinements):

DOF convergence graphs:

CPU convergence graphs:

## hp-FEM with iso, h-aniso and hp-aniso refinements¶

Final mesh (hp-FEM, isotropic refinements):

Final mesh (hp-FEM, h-anisotropic refinements):

Final mesh (hp-FEM, hp-anisotropic refinements):

DOF convergence graphs:

CPU convergence graphs: