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Anomalous diffusions are ubiquitous in nature, whose functional distributionsare governed by the backward Feynman-Kac equation. In this paper, the localdiscontinuous Galerkin (LDG) method is used to solve the 2D backwardFeynman-Kac equation in a rectangular domain. The spatial semi-discrete LDGscheme of the equivalent form (obtained by Laplace transform) of the originalequation is established. After discussing the properties of the fractionalsubstantial calculus, the stability and optimal convergence rates $O(h^{k+1})$of the semi-discrete scheme are proved by choosing an appropriate generalizednumerical flux. The $L1$ scheme on the graded meshes is used to deal with theweak singularity of the solution near the initial time. Based on thetheoretical results of a semi-discrete scheme, we investigate the stability andconvergence of the fully discrete scheme, which shows the optimal convergencerates $O(h^{k+1}+\tau^{\min\{2-\alpha,\gamma\delta\}})$. Numerical experimentsare carried out to show the efficiency and accuracy of the proposed scheme. Inaddition, we also verify the effect of the central numerical flux on theconvergence rates and the condition number of the coefficient matrix.

Local discontinuous Galerkin method for the Backward Feynman-Kac Equation