An analysis of galerkin proper orthogonal decomposition for subdiffusion

¹ Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK.
b.jin@ucl.ac.uk; bangti.jin@gmail.com
² Department of Applied Physics and Applied Mathematics, Columbia University, 10027 New York, USA.
zhizhou0125@gmail.com

Received: 25 August 2015
Revised: 26 January 2016
Accepted: 8 March 2016

Abstract

In this work, we develop a novel Galerkin-L1-POD scheme for the subdiffusion model with a Caputo fractional derivative of order α ∈ (0,1) in time, which is often used to describe anomalous diffusion processes in heterogeneous media. The nonlocality of the fractional derivative requires storing all the solutions from time zero. The proposed scheme is based on continuous piecewise linear finite elements, L1 time stepping, and proper orthogonal decomposition (POD). By constructing an effective reduced-order model using problem-adapted basis functions, it can significantly reduce the computational complexity and storage requirement. We shall provide a complete error analysis of the scheme under realistic regularity assumptions by means of a novel energy argument. Extensive numerical experiments are presented to verify the convergence analysis and the efficiency of the proposed scheme.