This problem is a coupled system of two equations with a mixed derivative in the coupling term (Lame equations); the context of the problem comes from the subject of linear elasticity.

Equation solved: Coupled system of two equations

where , and are the and displacements, is Young’s Modulus, and is Poisson’s ratio.

Domain of interest: with a slit from to .

Boundary conditions: Dirichlet, given by exact solution.

Known exact solution for mode 1:

here lambda = 0.5444837367825, and Q = 0.5430755788367.

Known exact solution for mode 2:

here lambda = 0.9085291898461, and Q = -0.2189232362488. Both in mode 1 and mode 2, , and .

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:

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

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

DOF convergence graphs:

CPU convergence graphs: