Error Estimates for Backward Fractional Feynman–Kac Equation with Non-Smooth Initial Data

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.

A Derivation of (4.11)

Without loss of generality, we denote $\varOmega =[0,1]$, $0=x_0<x_1<\cdots <x_n=1$, and $\varDelta x_i=x_{i}-x_{i-1}$. Let $\theta =\sup _{x\in \varOmega }\arg ((\beta _{\tau ,1}(z,x))^{\alpha })- \inf _{x\in \varOmega }\arg ((\beta _{\tau ,1}(z,x))^{\alpha })$ and we take $\kappa$ large enough to guarantee $\theta <\pi$. Then we have

$$\begin{aligned} \begin{aligned} \left| ((\beta _{\tau ,1}(z,x))^{\alpha }e,e) \right| =\left| \int _{0}^{1}(\beta _{\tau ,1}(z,x))^\alpha e(x)^2dx\right|&= \lim _{n\rightarrow \infty }\left| \sum _{i=1}^{n}\varDelta x_i(\beta _{\tau ,1}(z,x_i))^\alpha e(x_i)^2\right| \\ \ge&\cos \left( \frac{\theta }{2}\right) \lim _{n\rightarrow \infty }\sum _{i=1}^{n}\varDelta x_i e(x_i)^2 |(\beta _{\tau ,1}(z,x_i))^\alpha |\\ \ge&\,C\cos \left( \frac{\theta }{2}\right) \lim _{n\rightarrow \infty }\sum _{i=1}^{n}\varDelta x_i e(x_i)^2 |\beta _{\tau ,min}^\alpha |\\ \ge&\,C\cos \left( \frac{\theta }{2}\right) |\beta _{\tau ,min}^\alpha |\Vert e\Vert ^2_{L^2(\varOmega )}. \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \cos \left( \frac{\theta }{2}\right) |\beta _{\tau ,min}^\alpha |\Vert e\Vert ^2_{L^2(\varOmega )}\le C\left| ((\beta _{\tau ,1}(z,x))^{\alpha }e,e) \right| . \end{aligned}$$