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High-order BDF fully discrete scheme for backward fractional Feynman-Kac equation with nonsmooth data
The Feynman-Kac equation governs the distribution of the statistical
observable -- functional, having wide applications in almost all disciplines.
After overcoming challenges from the time-space coupled nonlocal operator and
the possible low regularity of functional, this paper develops the high-order
fully discrete scheme for the backward fractional Feynman-Kac equation by using
backward difference formulas (BDF) convolution quadrature in time, finite
element method in space, and some correction terms. With a systematic
correction, the high convergence order is achieved up to in time, without
deteriorating the optimal convergence in space and without the regularity
requirement on the solution. Finally, the extensive numerical experiments
validate the effectiveness of the high-order schemes.Comment: 23 page