Elliptic Eigenvalue Problems

Abstract

If $(L-\lambda I)u\;=\;f$ is not solvable, the Riesz–Schauder theory states that there are eigenvalues and eigenvectors. The weak formulation of the eigenvalue problem and some basic terms are discussed in Section 11.1. Since we do not require the system to be symmetric, also the adjoint problem must be treated. Section 11.2 is devoted to the finite-element discretisation by a family $\{V_h\;:\;h \in\;H\}$ of subspaces. Theorems 11.13 and 11.15 state an important result: Each eigenvalue λ ₀ of L is associated to a sequence of discrete eigenvalues converging to λ ₀, and vice versa. The corresponding error estimates are given in §11.2.3 for the case of simple eigenvalues. A related estimate for the eigenfunctions is provided by Lemma 11.23. Finally, Theorem 11.24 presents an improved error estimate of the eigenvalues by means of the eigenfunctions. The Riesz–Schauder theory also states that the equation $(L-\lambda I)u\;=\;f$> can even be solved for eigenvalues λ if f satisfies suitable side conditions. These equations are treated in §11.2.4. Section 11.3 discusses the discretisation by difference schemes. Also in this case similar results can be obtained.