NUMERICAL SOLUTION OF MULTIDIMENSIONAL SPECTRAL FRACTIONAL DIFFUSION PROBLEMS: FROM CAFFARELLI TO BURA

Lecturer: Prof. Sv. Margenov (IICT – BAS)
Time: 3:00 pm
Date: May 23, 2023
Place: Room 503 of IMI – BAS

Abstract:

Fractional diffusion operators appear naturally in many areas in mathematics, physics, ect. The most important property of the related b.v. problems is that they are nonlocal.

Let us consider the fractional power of a self-adjoint elliptic operator introduced through its spectral decomposition. It is also self-adjoint but nonlocal. Advanced numerical methods in this area have been heavily influenced by the pioneering work in differential operator theory by Caffarelli and Silvestre, “An Extension Problem Associated with the Fractional Laplacian”, 2007.

After discretization, nonlocal problems lead to linear systems with dense matrices. In the multidimensional case and domains with general geometry, the considered problems are extremely expensive from a computational point of view. Over the past decade, several different techniques have been proposed to localize the nonlocal operator, thereby increasing the spatial dimension of the computational domain.

We have developed an alternative approach. Let A be a SPD sparse matrix arising from finite element method (FEM) or finite difference method (FDM) discretization of the initial (local) problem.

Based on the best uniform rational approximations (BURA) of degree k of z^α, 0 ≤ z ≤ 1, a class of efficient solution methods for algebraic systems involving A^α, 0 < α < 1, is proposed and analysed. Robust error estimates with respect to the condition number κ(A) are derived, showing the exponential convergence of the BURA methods with respect to the degree of rational approximation.

Although the fractional power of A is a dense matrix, the algorithm has complexity of order O(N log²N), where N is the number of unknowns. At this point, we assume that some solver of optimal complexity (say multigrid or multilevel) is used for the auxiliary systems with matrices A + d_jI, d_j ≥ 0, j = 1, . . . , k.

The presented (up to 3D) numerical tests are focussed on problems with low regularity of the solutions, including cases of adaptive mesh refinement. The comparative analysis demonstrates the superiority of the BURA methods provided with rigorous theoretical results. Some recent works about BURA based preconditioning of coupled problems and non-overlapping domain decomposition methods are discussed at the end.

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