Error estimates for backward fractional Feynman-Kac equation with non-smooth initial data

TL;DR

This paper develops and analyzes numerical methods for the backward fractional Feynman-Kac equation with non-smooth initial data, achieving optimal convergence without regularity assumptions and validating with numerical experiments.

Contribution

It introduces a new discretization approach that does not require regularity assumptions, providing error estimates for both semi-discrete and fully discrete schemes.

Findings

First- and second-order convergence in time achieved.

Optimal convergence rates for finite element discretization.

Numerical experiments confirm the effectiveness of the methods.

Abstract

In this paper, we are concerned with the numerical solution for the backward fractional Feynman-Kac equation with non-smooth initial data. Here we first provide the regularity estimate of the solution. And then we use the backward Euler and second-order backward difference convolution quadratures to approximate the Riemann-Liouville fractional substantial derivative and get the first- and second-order convergence in time. The finite element method is used to discretize the Laplace operator with the optimal convergence rates. Compared with the previous works for the backward fractional Feynman-Kac equation, the main advantage of the current discretization is that we don't need the assumption on the regularity of the solution in temporal and spatial directions. Moreover, the error estimates of the time semi-discrete schemes and the fully discrete schemes are also provided. Finally, we perform the numerical experiments to verify the effectiveness of the presented algorithms.