On the Application of a Hierarchically Semi-separable Compression for Space-Fractional Parabolic Problems with Varying Time Steps

Abstract

Anomalous (fractional) diffusion is observed when the Brownian motion hypotheses are violated. It is modeled with the fractional Laplace operator, which can be defined in several ways. In this work we use the integral definition with the Riesz potential. For the discretization in space we apply the finite element method and for the discretization in time – a backward Euler scheme with varying time steps. The fractional Laplacian is a non-local operator and the arising stiffness matrix is dense. The time dependent problem is reduced to solving a sequence of linear systems whose matrices are constructed from the stiffness matrix, lumped mass matrix and the time step. When the time step changes we must refactorize the matrix before solving the current system. If the time step doesn’t change we can solve with the matrix factorized on a previous time step change. When utilizing the generic method using a block LU factorization, the computational complexity of the forward elimination is $$O(n^3)$$ and $$O(n^2)$$ of the backward substitution. In this work we develop an alternative method based on the hierarchically semi-separable (HSS) compression. With this method we compress the matrix at the beginning only. The HSS compression has a computational complexity $$O(n^2r)$$ . Then, when the time step changes we need to apply ULV-like factorization with computational complexity of $$O(nr^2)$$ . The solution step with the factorized matrix at each time step has computational complexity of O(nr). Here, r is the maximum off-diagonal rank of the approximate matrix, which is computed during the compression process. For suitable problems r is much smaller than the number of unknowns n. The numerical experiments presented show the advantages of the developed HSS compression based solution method.