Eigenvalue enclosures and convergence for the linearized MHD operator

Abstract

We discuss how to compute certified enclosures for the eigenvalues of benchmark linear magnetohydrodynamics (MHD) operators in the plane slab and cylindrical pinch configurations. For the plane slab, our method relies upon the formulation of an eigenvalue problem associated to the Schur complement, leading to highly accurate upper bounds for the eigenvalue. For the cylindrical configuration, a direct application of this formulation is possible, however, it cannot be rigourously justified. Therefore in this case we rely on a specialized technique based on a method proposed by Zimmermann and Mertins. In turns this technique is also applicable for finding accurate complementary bounds in the case of the plane slab. We establish convergence rates for both approaches.

On the quality of complementary bounds for eigenvalues

“…In the numerical experiments of this section we have "abused" slightly the notation and written the index as j for the eigenvalues of both H har and H anh . Figure 1 shows n versus the size of the eigenvalue enclosure λj,up − λ j,low , for λj,up > λj an upper bound found from (7) and λ j,low < λj a lower bound found from (6). In this figure the slopes are fairly close to 4, indicating that the convergence rates established in Theorem 5.3 are optimal.…”

Spectral Enclosure and Superconvergence for Eigenvalues in Gaps

“…We also note the relatively poor performance of the quadratic methods. The latter is not entirely surprising as the known convergence rates for quadratic methods are measured in terms of δ A (L({λ}), L n ), i.e., the distance of the eigenspace to the trial space with respect to the graph norm; see [8,Lemma 2] and [28, Section 6].…”

Analysis of an Unconditionally Convergent Stabilized Finite Element Formulation for Incompressible Magnetohydrodynamics

“…Remark 5.1 It is not the target of the method developed in this work the eigenvalue MHD problem [14]. Since the requirements for a method to be useful for initial and boundary value problems, viz.…”