A new mixed finite-element method for $H^2$ elliptic problems
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.
Cite
@article{arxiv.2105.07289,
title = {A new mixed finite-element method for $H^2$ elliptic problems},
author = {Patrick E. Farrell and Abdalaziz Hamdan and Scott P. MacLachlan},
journal= {arXiv preprint arXiv:2105.07289},
year = {2022}
}