Ritz value estimates and applications in mathematical physics

A major contribution of this thesis is a (new) general theory on Ritz value spectral approximations for elliptic self-adjoint operators. These results extend the known Davis--Kahan estimates to operators defined as quadratic forms. As a first application, we have considered the problem of computing rigorous spectral estimates for problems of the large coupling limit (for both eigenvalues and spectral families). In this context two problem classes have been identified: the regular case which encompasses problems from the elasticity theory (Arch/Curved Rod, Lame/Stokes) and the second class of problems (Schroedinger/Dirichlet, Helmholtz/Dirichlet) which does not satisfy the new regularity requirement. As a second class of problems we have studied the spectral finite element approximations for operators which are not H2 regular.