Approximation of variational eigenvalue problems
The eigenvalue problem to be approximated is \[ a(u,v)=\lambda b(u,v) \quad \forall v\in V; \] \(V\) is a Hilbert space \(a,b\) are bilinear forms, \(\lambda\in \mathbb{R}\), \(u\in V\backslash \{0\}\). The variational properties of the eigenvalues and eigenfunctions are determined by the Rayleigh functional \(R(v)=a(v,v)/b(v,v)\) for \(v\in V\backslash \{0\}.\) The original infinite-dimensional problem is approximated by the finite-dimensional one: Find \(\lambda^{h}\in \mathbb{R}\) and \(u^{h}\in V_{h}\backslash \{0\}\) such that: \[ a_{h}(u^{h},v^{h})=\lambda^{h} b_{h}(u^{h},v^{h}) \quad \forall v^{h}\in V_{h}. \] Here \(V_{h}\) has the finite dimension \(N_{h}\rightarrow\infty\) as \(h\rightarrow 0.\) The discrete Rayleigh functional is given by the expression \[ R_{h}(v^{h})=a_{h}(v^{h},v^{h})/b_{h}(v^{h},v^{h}) \quad\forall v^{h}\in V_{h}\backslash \{0\}. \] The first three theorems refer to the approximate eigenvalues \(0<\lambda^{h}_{1}\leq\lambda^{h}_{2}\leq\cdot\cdot\cdot\leq\lambda^{h}_{N_{n}},\) and the corresponding \(\mathcal{E}_{k}(W_{h})\), denoting the set of all \textit{k-}dimensional subsets of \(W_{h}\). Set \(\mathcal{E}_{k}^{h}=\mathcal{E}_{k}(V_{h}). \) One of these theorems asserts that \[ \begin{multlined} \lambda^{h}_{k}=\underset{W_{h}\in\mathcal{E}_{k-1}^{h}}{\max}\underset {v^{h} \in (W_{h})_{\alpha_{h}}^{\bot}\backslash \{0\}}{\min}R_{h}(v^{h})=\underset{W_{h}\in\mathcal{E}_{N_{h}-k+1}^{h}}{\max}\underset {v^{h}\in W_{h}\backslash \{0\}}{\min}R_{h}(v^{h})=\\ \underset{W_{h}\in\mathcal{E}_{N_{h}-k+1}^{h}}{\min}\underset {v^{h}\in W_{h}\backslash \{0\}}{\max}R_{h}(v^{h}).\end{multlined} \] For the estimate of the error of the approximate eigenvalues the author gets the following inequality: \[ |\lambda_{k}^{h}-\lambda_{k}|\leq c[\delta^{h}_{2}+(\varepsilon^{h}+\delta_{1}^{h})^{2}] \] for all \(h\) sufficiently small. As an example the author considers the boundary value problem: \[ -(pu')'+qu=\lambda ru, \quad x\in(0,l) \quad u(0)=u(l)=0. \]
scientific article; zbMATH DE number 5824153