Demo: The 2-D Steklov-Poincaré problem: P₀ Boundary Element

In this demo, we consider a problem derived from the Laplace equation, namely the Steklov-Poincaré problem. The problem is the following: Given a function g defined on the boundary of the domain. We look for the normal derivative of the solution of the Laplace equation whose trace on the boundary is g. The problem is solved by an integral representation by using P₀ boundary elements. In the OFELI library contains the class SteklovPoincare2DBE for such a purpose.

• The code starts like the previous ones concerning the Laplace equation. Then, we declare two vectors: The vector g will store the prescribed boundary value, and the vector g, will contain the solution i.e. the normal derivative. This is why it is declared as a BOUNDARY_SIDE_FIELD, i.e. constant on each boundary side.
  

  Vect g(ms), u(ms,0,BOUNDARY_SIDE_FIELD);

• As an example, we consider an annular domain defined by the intersection of the circle with center (0,0) and radius 2, with the circle with center (0,0) and radius 1. We impose in this example, the values g=0 on the inner boundary and g=1 on the outer boundary.
  

  g.setNodeBC(1,"0."); g.setNodeBC(2,"1.");

• We can now instantiate the class SteklovPoincare2DBE. We also transmit the boundary condition to the equation using the member function setInput
  

  SteklovPoincare2DBE eq(ms,u); eq.setInput(BOUNDARY_CONDITION,g);

• We can now solve the problem and print out the solution. We also compute the L₂-norm of the solution to check convergence.

  eq.run(); cout << u; cout << "L2 solution norm: " << u.Norm(WNORM2) << endl;