Numerical approximation of the spectrum of self-adjoint operators in operator preconditioning

We consider operator preconditioning ${\mathcal{B}}^{-1}\mathcal{A}$, which is employed in the numerical solution of boundary value problems. Here, the self-adjoint operators $\mathcal{A},\mathcal{B}:H_0^1(\mathrm{\Omega })\to H^{-1}(\mathrm{\Omega })$ are the standard integral/functional representations of the partial differential operators −∇⋅ (k(x)∇u) and −∇⋅ (g(x)∇u), respectively, and the scalar coefficient functions k(x) and g(x) are assumed to be continuous throughout the closure of the solution domain. The function g(x) is also assumed to be uniformly positive. When the discretized problem, with the preconditioned operator ${\mathcal{B}}_n^{-1}{\mathcal{A}}_n$, is solved with Krylov subspace methods, the convergence behavior depends on the distribution of the eigenvalues. Therefore, it is crucial to understand how the eigenvalues of ${\mathcal{B}}_n^{-1}{\mathcal{A}}_n$ are related to the spectrum of ${\mathcal{B}}^{-1}\mathcal{A}$. Following the path started in the two recent papers published in SIAM J. Numer. Anal. [57 (2019), pp. 1369-1394 and 58 (2020), pp. 2193-2211], the first part of this paper addresses the open question concerning the distribution of the eigenvalues of ${\mathcal{B}}_n^{-1}{\mathcal{A}}_n$ formulated at the end of the second paper. The approximation of the spectrum studied in the present paper differs from the eigenvalue problem studied in the classical PDE literature which addresses individual eigenvalues of compact (solution) operators.

In the second part of this paper, we generalize some of our results to general bounded and self-adjoint operators $\mathcal{A},B:V\to V^{\#}$, where V^# denotes the dual of V. More specifically, provided that $\mathcal{B}$ is coercive and that the standard Galerkin discretization approximation properties hold, we prove that the whole spectrum of ${\mathcal{B}}^{-1}\mathcal{A}:V\to V$ is approximated to an arbitrary accuracy by the eigenvalues of its finite dimensional discretization ${\mathcal{B}}_n^{-1}{\mathcal{A}}_n$.