Analysis of an Unconditionally Convergent Stabilized Finite Element Formulation for Incompressible Magnetohydrodynamics

Badia, Santiago; Codina, Ramon; Planas, Ramón

doi:10.1007/s11831-014-9129-5

Cited by 20 publications

(15 citation statements)

References 45 publications

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“…FEMPAR has already been successfully used in a wide set of applications by the authors of the library: simulation of turbulent flows and stabilized FE methods [ 46 – 49 ], magnetohydrodynamics [ 50 – 54 ], monotonic FEs [ 55 – 59 ], unfitted FEs and embedded boundary methods [ 60 ], and additive manufacturing simulations [ 61 ]. It has also been used for the highly efficient implementation of DD solvers [ 34 , 37 , 39 , 62 – 66 ] and block preconditioning techniques [ 44 ].…”

Section: The Fempar Projectmentioning

confidence: 99%

FEMPAR: An Object-Oriented Parallel Finite Element Framework

Badia

Martı́n

Principe

2017

Arch Computat Methods Eng

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129

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FEMPAR is an open source object oriented Fortran200X scientific software library for the high-performance scalable simulation of complex multiphysics problems governed by partial differential equations at large scales, by exploiting state-of-the-art supercomputing resources. It is a highly modularized, flexible, and extensible library, that provides a set of modules that can be combined to carry out the different steps of the simulation pipeline. FEMPAR includes a rich set of algorithms for the discretization step, namely (arbitrary-order) grad, div, and curl-conforming finite element methods, discontinuous Galerkin methods, B-splines, and unfitted finite element techniques on cut cells, combined with h-adaptivity. The linear solver module relies on state-of-the-art bulk-asynchronous implementations of multilevel domain decomposition solvers for the different discretization alternatives and block-preconditioning techniques for multiphysics problems. FEMPAR is a framework that provides users with out-of-the-box state-of-the-art discretization techniques and highly scalable solvers for the simulation of complex applications, hiding the dramatic complexity of the underlying algorithms. But it is also a framework for researchers that want to experience with new algorithms and solvers, by providing a highly extensible framework. In this work, the first one in a series of articles about FEMPAR, we provide a detailed introduction to the software abstractions used in the discretization module and the related geometrical module. We also provide some ingredients about the assembly of linear systems arising from finite element discretizations, but the software design of complex scalable multilevel solvers is postponed to a subsequent work.

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Section: The Fempar Projectmentioning

confidence: 99%

FEMPAR: An Object-Oriented Parallel Finite Element Framework

Badia

Martı́n

Principe

2017

Arch Computat Methods Eng

Self Cite

129

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“…While there has been extensive research and literature on the Navier‐Stokes subproblem and ,() the extended magnetic induction subproblem and has been mainly studied in context of the magnetohydrodynamical problem and () and seldom for itself . For that reason, we mainly focus on and and propose two stabilized Lagrangian nodal‐based magnetic diffusivity finite element methods with additional stabilization terms.…”

Section: Introductionmentioning

confidence: 99%

Two variants of magnetic diffusivity stabilized finite element methods for the magnetic induction equation

Wacker

2019

Math Methods in App Sciences

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We consider the time-dependent magnetic induction model where the sought magnetic field interacts with a prescribed velocity field. This coupling results in an additional force term and time dependence in Maxwell's equation. We propose two different magnetic diffusivity stabilized continuous nodal-based finite element methods for this problem. The first formulation simply adds artificial magnetic diffusivity to the partial differential equation, whereas the second one uses a local projected magnetic diffusivity as stabilization. We describe those methods and analyze them semi-discretized in space to get bounds on stabilization parameters where we distinguish equal-order elements and Taylor-Hood elements. Different numerical experiments are performed to illustrate our theoretical findings. KEYWORDS layer phenomena, local projection stabilization, magnetic diffusivity, magnetic induction equation, stabilized finite element methods, stability and convergence analysis 4554 4555 State-of-the-art methodsBefore stating our contributions, we discuss methods mainly used in computational electrodynamics. Curl-conforming finite element methods are the standard approach for approximating (3) and (4). 9,13-16 Although these elements work well with the problem's structure, they have some drawbacks in implementation. One important disadvantage is the occurrence of ill-posed matrices. In problems with many degrees of freedom, the corresponding problem's matrix can even be singular. The introduction of Lagrangian multipliers improves this situation. However, this increases the number of unknowns and can cause trouble in calculations. 17 Because of their efficiency and their ease in implementation, nodal-based finite element methods seem to be an attractive alternative.Costabel and Dauge proposed a weighted regularization for Maxwell's equation in polyhedral domains with reentrant corners in 2002. 18 This article caused a rehabilitation of nodal-based finite element methods in computational electromagnetism. Follow-up works suggested different regularizations for Maxwell's problem. [19][20][21] These ideas led to different stabilization techniques in case of magnetohydrodynamics. 7,8,11 ContributionsOur starting point is the stabilization method in Kaya et al. 12 It stabilizes the divergence constraint and the gradient of the magnetic pseudo-pressure. Although this algorithm works well for moderately small magnetic diffusivity , our work will demonstrate that it is not well suited for nearly vanishing . This observation motivates our following contributions: 1) We first expand the aforementioned method by a local projection stabilization of the induction term ∇ × (u × b) as solely done in numerical simulations in Bonito et al. 22 Further and in contrast with Kaya et al 12 and Bonito et al, 22 we stabilize the magnetic diffusion term in two possible ways. We either simply add artificial magnetic diffusivity or local projected magnetic diffusivity. These ideas are inspired by stabilization techniques from turbulence modeling. 5,23...

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“…in . Extensions of nodal‐based stabilized FEM to the resistive MHD model can be found in . The focus of the latter papers is on the case of equal‐order interpolation of all unknown variables.…”

Section: Introductionmentioning

confidence: 99%

“…Stabilization techniques of residual type based on nodal-based FEM were considered for the Maxwell problem by Codina/Badia in [4] and Bonito et al in [5,6]. Extensions of nodal-based stabilized FEM to the resistive MHD model can be found in [7][8][9]. The focus of the latter papers is on the case of equal-order interpolation of all unknown variables.…”

Section: Introductionmentioning

confidence: 99%