The multigrid method of Wilson element on arbitrary quadrilateral meshes with two new variational formulations

1998 ◽  
Vol 14 (2) ◽  
pp. 134-139 ◽  
Author(s):  
Ma Liming ◽  
Chang Qianshun
Keyword(s):  
Multigrid Method ◽  
Quadrilateral Meshes ◽  
Wilson Element ◽  
Variational Formulations

Convergence of the nonconforming Wilson element for arbitrary quadrilateral meshes

1980 ◽  
Vol 36 (1) ◽  
pp. 33-52 ◽  
Author(s):  
P. Lesaint ◽  
M. Zl�mal
Keyword(s):  
Quadrilateral Meshes ◽  
Wilson Element

scholarly journals Variational formulations of steady rotational equatorial waves

2020 ◽  
Vol 10 (1) ◽  
pp. 534-547
Author(s):  
Jifeng Chu ◽  
Joachim Escher
Keyword(s):  
Linear Stability ◽  
Stream Function ◽  
Water Waves ◽  
Variational Formulation ◽  
Energy Functional ◽  
Equatorial Waves ◽  
Second Variation ◽  
Governing Equations ◽  
The Second Variation ◽  
Variational Formulations

Abstract When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f-plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional 𝓗 in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves.

scholarly journals A Flexible, Parallel, Adaptive Geometric Multigrid Method for FEM

2021 ◽  
Vol 47 (1) ◽  
pp. 1-27
Author(s):  
Thomas C. Clevenger ◽  
Timo Heister ◽  
Guido Kanschat ◽  
Martin Kronbichler
Keyword(s):  
Multigrid Method ◽  
Geometric Multigrid Method ◽  
Geometric Multigrid

scholarly journals fenicsR13

2021 ◽  
Vol 47 (2) ◽  
pp. 1-29
Author(s):  
Lambert Theisen ◽  
Manuel Torrilhon
Keyword(s):  
Gas Flow ◽  
Differential Operators ◽  
Mixed Finite Element ◽  
Test Cases ◽  
Typical Application ◽  
Flow Models ◽  
Computing Platform ◽  
Abstraction Levels ◽  
Variational Formulations ◽  
Rarefaction Effects

We present a mixed finite element solver for the linearized regularized 13-moment equations of non-equilibrium gas dynamics. The Python implementation builds upon the software tools provided by the FEniCS computing platform. We describe a new tensorial approach utilizing the extension capabilities of FEniCS’ Unified Form Language to define required differential operators for tensors above second degree. The presented solver serves as an example for implementing tensorial variational formulations in FEniCS, for which the documentation and literature seem to be very sparse. Using the software abstraction levels provided by the Unified Form Language allows an almost one-to-one correspondence between the underlying mathematics and the resulting source code. Test cases support the correctness of the proposed method using validation with exact solutions. To justify the usage of extended gas flow models, we discuss typical application cases involving rarefaction effects. We provide the documented and validated solver publicly.

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