Two Stabilized Three-Field Formulations for the Biharmonic Problem

Banz, Lothar; Petsche, Jan; Schröder, Jörg

doi:10.1007/978-3-030-14244-5_3

Cited by 2 publications

(2 citation statements)

References 19 publications

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“…These share the disadvantages of C0IP methods, while requiring more degrees of freedom than 𝐶 0 approaches. Another attractive option to avoid using 𝐻 2 -conforming methods is mixed finite-element methods, in which the gradient or the Laplacian of the solution are approximated in addition to the solution itself [4,[16][17][18][19][20][21][22][23][24][25]. A natural classification of such mixed finite-element methods is based on how many functions (fields) are directly approximated.…”

Section: Introductionmentioning

confidence: 99%

A new mixed finite-element method for H2 elliptic problems

Farrell

Abdalaziz

MacLachlan

2022

Computers & Mathematics with Applications

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Section: Introductionmentioning

confidence: 99%

A new mixed finite-element method for H2 elliptic problems

Farrell

Abdalaziz

MacLachlan

2022

Computers & Mathematics with Applications

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“…Another attractive option to avoid using H 2 -conforming methods is mixed finiteelement methods, in which the gradient or the Laplacian of the solution are approximated in addition to the solution itself [13][14][15]23,24,40,42]. A natural classification of such mixed finite-element methods is based on how many functions (fields) are directly approximated.…”

mentioning

confidence: 99%

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

Farrell¹,

Abdalaziz²,

MacLachlan³

2021

Preprint

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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 finiteelement 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.

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Two Stabilized Three-Field Formulations for the Biharmonic Problem

Cited by 2 publications

References 19 publications

A new mixed finite-element method for H2 elliptic problems

A new mixed finite-element method for H2 elliptic problems

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

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