Semi and Fully Discrete Analysis of Extended Fisher–Kolmogorov Equation with Nonstandard FEMs for Space Discretisation - Das, Avijit; Nataraj, Neela; Chirappurathu Remesan, Gopikrishnan (Journal of scientific computing, 2025)
This article discusses lowest-order nonstandard finite element methods for space discretisation and backward Euler scheme for time discretisation of the extended Fisher–Kolmogorov equation with clamped boundary conditions. Spatial discretisation employs popular piecewise quadratic schemes based on triangles, namely, the Morley, the discontinuous Galerkin, and the C 0 interior penalty schemes. Based on the smoother J I M defined for a piecewise smooth input function by a (generalized) Morley interpolation I M followed by a companion operator J from Carstensen and Nataraj (ESAIM Math Model Numer Anal 56(1):41–78, 2022), a smoother based Ritz projection operator is defined. A set of abstract hypotheses establish the approximation properties of the Ritz projection operator. The approach allows for an elegant semidiscrete and fully discrete error analysis with minimal regularity assumption on the exact solution. Error estimates for both the semidiscrete and fully discrete schemes are presented. The numerical results validate the theoretical estimates and demonstrate the capability of the discontinuous Galerkin method to approximate the solution, even for non-smooth initial condition.