Two-grid methods for a new mixed finite element approximation of semilinear parabolic integro-differential equations
C. Liu1, T. Hou2
1Institute of Computational Mathematics, Yongzhou 425100, Hunan, China
2School of Mathematics and Statistics, 132013, Jilin, China
Keywords: полулинейные параболические интегро-дифференциальные уравнения, новый смешанный метод конечных элементов, априорная оценка ошибки, двухсеточный, пространство квадратично интегрируемых функций, semilinear parabolic integro-differential equations, a new mixed finite element method, a priori error estimate, two-grid, space of square integrable functions
Abstract
In this paper, we present a two-grid scheme for a semilinear parabolic integro-differential equation using a new mixed finite element method. The gradient for the method belongs to the space of square integrable functions instead of the classical H (div;Ω) space. The velocity and the pressure are approximated by a P 02- P 1 pair which satisfies an inf-sup condition. Firstly, we solve the original nonlinear problem on the coarse grid in our two-grid scheme. Then, to linearize the discretized equations, we use Newton's iteration on the fine grid twice. It is shown that the algorithm can achieve an asymptotically optimal approximation as long as the mesh sizes satisfy h = O ( H 6 |ln H |2). As a result, solving such a large class of nonlinear equations will not be much more difficult than solving one linearized equation. Finally, a numerical experiment is provided to verify the theoretical results of the two-grid method.
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