A space-time finite element method for solving linear Riesz space fractional partial differential equations

In this paper, numerical solutions for linear Riesz space fractional partial differential equations with a second-order time derivative are considered. A space-time finite element method is proposed to solve these equations numerically. In the time direction, the C-0-continuous Galerkin method is used to approximate the second-order time derivative. In the space direction, the usual linear finite element method is developed to approximate the space fractional derivative. The matrix equivalent form of this numerical method is deduced. The stability of the discrete solution is established and the optimal error estimates are investigated. Some numerical tests are given to validate the theoretical results.

Automated summary

This paper discusses numerical solutions for linear Riesz space fractional partial differential equations with a second-order time derivative, proposing a space-time finite element method for numerical solutions. The method approximates the second-order time derivative using the C-0-continuous Galerkin method in the time direction and develops the usual linear finite element method to approximate the space fractional derivative in the space direction. The stability of the discrete solution is established, optimal error estimates are investigated, and numerical tests are conducted to validate the theoretical results.