A new mixed finite-element method for $H^2$ elliptic problems

TL;DR

This paper introduces a novel three-field mixed finite-element method for fourth-order elliptic problems, effectively handling boundary conditions and providing efficient solvers, with demonstrated accuracy and computational efficiency in 2D and 3D.

Contribution

It presents a new mixed finite-element formulation for fourth-order problems that incorporates boundary conditions weakly and develops multigrid solvers for the resulting saddle-point systems.

Findings

The method achieves optimal accuracy in numerical tests.

Multigrid solvers significantly improve computational efficiency.

The formulation effectively handles natural boundary conditions.

Abstract

Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finite-element discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed.