Objectives: To implement the GMRES algorithm and to solve the exterior Dirichlet problem for Laplace's equation using a boundary integral equation.

(1)

Replace the linear system solver in your code from Assignment 1 to use the GMRES algorithm (if you are not already using this). Report on how the number of GMRES iterations is affected by the mesh size (for a fixed ellipse) and how it is affected by the eccentricity of the ellipse.

(2)

Convert your code from Assignment 1 (with GMRES as the linear system solver) to solve the exterior Dirichlet problem: Du = 0 outside the ellipse

x2 a2 + y2 b2 = 1 ,

for given Dirichlet data prescribed on the boundary. An appropriate integral equation for this problem is:

- m( ^ Q   ) + 1 p ó õ S  m(Q) é ê ë ¶ ¶nQ æ è log| Q - ^ Q   | ö ø + 1 ù ú û  ds(Q) = 2 f( ^ Q   ) ,    ^ Q   Î S ,

where m(Q) is the jump in the solution u at the point Q. The solution at any point P outside the ellipse D is then given by:

u(P) = 1 2 p ó õ S  m(Q) é ê ë ¶ ¶nQ ( log| Q-P | ) + 1 ù ú û  ds(Q) .

Test your code by taking the exact solution to be

u(x,y) = 1 + 2 æ ç è x - x0 (x - x0 )2 + (y - y0 )2 ö ÷ ø

for some point ( x₀ , y₀ ) inside the ellipse. Once you have computed the density function m(Q), use the integral representation to evaluate u(P) at several points outside the ellipse. Again determine how the error decreases with the mesh spacing h between the boundary points and how the accuracy is affected by the eccentricity of the ellipse.

Use the integral representation for u(P) to evaluate the solution at points near the boundary of the ellipse and explain what you find.