Hybrid high-order method for the extended Fisher-Kolmogorov and the Fisher-Kolmogorov equations

This article analyzes hybrid high-order method for space discretization and backward Euler and Crank-Nicolson schemes for time discretization of the nonlinear extended Fisher-Kolmogorov and the Fisher-Kolmogorov equations. The critical parameter γ > 0 is incorporated in the stabilization term of the HHO method. Error estimate of order 𝒪 ( h k +1 + Δ t ) (resp. 𝒪 ( h k +1 + (Δ t ) 2 ) in the energy-norm for the backward Euler (resp. Crank-Nicolson ) scheme is obtained when polynomials of order k +2 (resp. k ) with k ≥ 0 are utilized to approximate the exact solution in the interior of the polygon and its traces on the boundary of the polygon (resp. normal derivative on the mesh faces). The HHO discretization for the Fisher-Kolmogorov equation, that is, when the parameter γ = 0, leads to a convergence rate of 𝒪 ( h k +2 ) in the space variable. The results of the numerical experiments validate the theoretical results.