A unified mixed finite element method for fourth-order time-dependent problems using biorthogonal systems

This article introduces a unified mixed finite element framework based on a saddle -point formulation that applies to time -dependent fourth order linear and nonlinear problems with clamped, simply supported, and Cahn -Hilliard type boundary conditions. The classical mixed formulations lead to large matrix systems that demand huge storage and computational time making the schemes expensive, especially for the time -dependent problems. The proposed scheme circumvents this by employing biorthogonal basis functions that lead to sparse and positivedefinite systems. The article discusses a mixed finite element method for the biharmonic problem and the time -dependent linear and nonlinear versions of the extended Fisher-Kolmogorov equations equipped with the aforementioned boundary conditions. The wellposedness of the scheme is discussed and a priori error estimates are presented for the semi -discrete and fully discrete finite element schemes. The numerical experiments validate the theoretical estimates derived in the paper.