Hierarchical Super-Localized Orthogonal Decomposition Method

TL;DR

This paper introduces a hierarchical, sparse, compressed operator for elliptic PDEs with rough coefficients, enabling efficient multilevel solutions with optimal error rates even in high-contrast scenarios.

Contribution

It develops a novel superlocalized orthogonal decomposition method that constructs a hierarchical basis for efficient PDE solution approximation.

Findings

Achieves mesh-independent condition numbers for linear systems at each hierarchy level.

Maintains optimal error rates with superlocalized basis functions in high-contrast coefficient cases.

Enables parallel solution of independent linear systems across hierarchy levels.

Abstract

We present the construction of a sparse-compressed operator that approximates the solution operator of elliptic PDEs with rough coefficients. To derive the compressed operator, we construct a hierarchical basis of an approximate solution space, with superlocalized basis functions that are quasi-orthogonal across hierarchy levels with respect to the inner product induced by the energy norm. The superlocalization is achieved through a novel variant of the Super-Localized Orthogonal Decomposition method that is built upon corrections of basis functions arising from the Localized Orthogonal Decomposition method. The hierarchical basis not only induces a sparse compression of the solution space but also enables an orthogonal multiresolution decomposition of the approximate solution operator, decoupling scales and solution contributions of each level of the hierarchy. With this decomposition, the solution of the PDE reduces to the solution of a set of independent linear systems per level with mesh-independent condition numbers that can be computed simultaneously. We present an accuracy study of the compressed solution operator as well as numerical results illustrating our theoretical findings and beyond, revealing that desired optimal error rates with well-behaved superlocalized basis functions can still be attained even in the challenging case of coefficients with high-contrast channels.