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Superclose estimates analysis of a new mixed finite elements method for generalized nerve conduction equation. (Chinese. English summary) Zbl 1399.65275

Summary: Based on the nonconforming \(EQ_1^{{\text{rot}}}\) element and the Raviart-Thomas(R-T) element, a new lower order nonconforming mixed finite elements method is proposed for generalized nerve conduction equation. Firstly, the existence and uniqueness of approximation solutions are proved. Secondly, based on the high accuracy results of the above two elements and derivative transferring technique with respect to the time variable, the superclose with order \(O\left( {{h^2}} \right)\) for the primitive solution in \({H^1}\)-norm and the intermediate variable \(p\) in \({L^2}\)-norm are obtained under semi-discrete scheme respectively. Finally, a new fully-discrete approximation scheme is proposed and the superclose estimates with order \(O\left( {{h^2} + {\tau ^2}} \right)\) are deduced for the primitive solution in \({H^1}\)-norm and the intermediate variable \(p\) in \({L^2}\)-norm respectively. Here, \(h\) and \(\tau \) are the subdivision parameters in space and time step respectively.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
92C20 Neural biology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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