NIST-06 (Boundary Layer)¶

Git reference: Benchmark nist-06.

This example is a singularly perturbed problem that exhibits a boundary layer along the right and top sides of the domain.

Model problem¶

Equation solved: Convection-diffusion equation with first order terms.

-\epsilon \nabla^{2} u + 2\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}= f.

Domain of interest: (-1, 1)^2.

Boundary conditions: Dirichlet, given by exact solution.

Exact solution¶

u(x,y) = (1 - e^{-(1 - x) / \epsilon})(1 - e^{-(1 - y) / \epsilon})cos(\pi (x + y))

where \epsilon determines the strength of the boundary layer.

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

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

  virtual double value(double x, double y) const{
    return -epsilon*(-2*pow(M_PI,2)*(1 - exp(-(1 - x)/epsilon))*(1 - exp(-(1 - y)/epsilon))*cos(M_PI*(x + y))
       + 2*M_PI*(1 - exp(-(1 - x)/epsilon))*exp(-(1 - y)/epsilon)*sin(M_PI*(x + y))/epsilon
       + 2*M_PI*(1 - exp(-(1 - y)/epsilon))*exp(-(1 - x)/epsilon)*sin(M_PI*(x + y))/epsilon
       - (1 - exp(-(1 - y)/epsilon))*cos(M_PI*(x + y))*exp(-(1 - x)/epsilon)/pow(epsilon,2)
       - (1 - exp(-(1 - x)/epsilon))*cos(M_PI*(x + y))*exp(-(1 - y)/epsilon)/pow(epsilon,2))
       - 3*M_PI*(1 - exp(-(1 - x)/epsilon))*(1 - exp(-(1 - y)/epsilon))*sin(M_PI*(x + y))
       - 2*(1 - exp(-(1 - y)/epsilon))*cos(M_PI*(x + y))*exp(-(1 - x)/epsilon)/epsilon
       - (1 - exp(-(1 - x)/epsilon))*cos(M_PI*(x + y))*exp(-(1 - y)/epsilon)/epsilon;
  }