A Mixed FEM for a Triharmonic Equation Using a Biorthogonal System

ABSTRACT This article presents a mixed finite element method (FEM) for solving a triharmonic problem with Cahn‐Hilliard boundary conditions. The mixed FEM is formulated as a saddle point problem, which allows the use of a biorthogonal basis system. This method reduces the computational cost of solving a sixth‐order problem to that of a second‐order Poisson problem. The approach demonstrates not only computational efficiency but also significant benefits in terms of storage, particularly for time‐dependent problems. The article discusses the well‐posedness, stability, and error estimates of the proposed scheme. The approach is also extended to a time‐dependent sixth‐order problem, which is a linearized phase field crystal (PFC) problem. The mass conservation and energy stability of the discrete formulation of the time‐dependent problem are shown. The algebraic formulation is provided along with numerical validation of the approach.