[Paper Review] An Analysis of Galerkin Proper Orthogonal Decomposition for Subdiffusion

Bangti Jin, Zhi Zhou|arXiv (Cornell University)|Aug 25, 2015

Fractional Differential Equations Solutions24 references3 citations

TL;DR

This paper introduces a Galerkin-L1-POD scheme for subdiffusion problems with a Caputo fractional derivative, combining finite elements, L1 time discretization, and proper orthogonal decomposition (POD) to reduce computational cost and storage. It provides a complete error analysis using a novel energy argument, proving optimal convergence rates under realistic regularity assumptions and demonstrating high efficiency through numerical experiments.

ABSTRACT

In this work, we develop a novel Galerkin-L1-POD scheme for the subdiffusion model with a Caputo fractional derivative of order $α\in (0,1)$ 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 scheme 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.

Motivation & Objective

• To address the high computational and storage costs of solving subdiffusion problems with Caputo fractional derivatives due to their nonlocal nature.
• To develop a reduced-order model using proper orthogonal decomposition (POD) tailored to the solution manifold of subdiffusion.
• To provide a complete error analysis for the proposed Galerkin-L1-POD scheme under realistic regularity assumptions.
• To demonstrate the efficiency and convergence of the scheme through extensive numerical experiments.

Proposed method

• The method combines continuous piecewise linear finite elements in space with L1 time stepping for the Caputo fractional derivative.
• It constructs snapshots from fully discrete solutions and fractional difference quotients to generate POD basis functions.
• A reduced-order Galerkin framework is applied using the first $ m $ POD basis functions to form a low-dimensional solution space $ X_h^m $.
• The error analysis employs a novel energy argument to derive optimal convergence rates under weak regularity assumptions.
• The scheme is validated through extensive numerical experiments on 1D, 2D, and 3D domains.

Experimental results

Research questions

• RQ1 Can a reduced-order Galerkin-L1-POD scheme be constructed for subdiffusion with a Caputo fractional derivative to significantly reduce computational cost?
• RQ2 What is the convergence behavior of the Galerkin-L1-POD scheme under realistic regularity assumptions on the solution?
• RQ3 How does the proposed scheme compare in efficiency and accuracy to standard full-order methods?
• RQ4 Can a novel energy argument be developed to rigorously analyze the error in the reduced-order scheme?

Key findings

• The proposed Galerkin-L1-POD scheme achieves optimal convergence rates in both space and time under realistic regularity assumptions, including limited smoothing due to the fractional derivative.
• The error analysis is based on a novel energy argument that accounts for the nonlocality and weak regularity of the subdiffusion solution.
• Numerical experiments confirm the theoretical convergence rates and demonstrate significant reductions in computational cost and storage requirements.
• The POD basis, derived from solution snapshots, captures the dominant modes of the solution manifold, enabling effective model reduction.
• The scheme maintains stability and accuracy even when the solution exhibits singularities in time derivatives, such as $ t^{eta-1} $ behavior with $ eta = eta $.