NIST-07 (Boundary Line Singularity)¶

Git reference: Benchmark nist-07.

Many papers on testing adaptive algorithms use a 1D example with a singularity of the form x^{\alpha} at the left endpoint of the domain. This can be extended to 2D by simply making the solution be constant in y.

Model problem¶

Equation solved: Poisson equation

(1)-\Delta u = f.

Domain of interest: Unit Square (0, 1)^2

Boundary conditions: Dirichlet, given by exact solution.

Exact solution¶

u(x,y) = x^{\alpha}

where \alpha \geq 0.5 determines the strength of the singularity.

Right-hand side: Obtained by inserting the exact solution into the equation. The corresponding code snippet is shown below:

{
public:
  CustomRightHandSide(double alpha) : DefaultNonConstRightHandSide(), alpha(alpha) {};

  virtual double value(double x, double y) const {
    return - alpha * (alpha - 1.) * pow(x, alpha - 2.);
}