A new mixed finite-element method for H2 elliptic problems. (15th December 2022)

• Record Type: Journal Article
• Authors: Farrell, Patrick E.; Hamdan, Abdalaziz; MacLachlan, Scott P.
• Is Part Of: Computers & mathematics with applications. Volume 128(2022)
• Page Start: 300
• Page End: 319
• Publication Date: 2022-12-15
• Subjects: Mixed finite-element methods -- Biharmonic equation -- Multigrid methods -- Saddle-point problems; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2022.10.024
• Languages: English
• ISSNs: 0898-1221

Abstract

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.