Analysis of new augmented Lagrangian formulations for mixed finite element schemes

1997 ◽  
Vol 75 (4) ◽  
pp. 405-419 ◽  
Author(s):  
Daniele Boffi ◽  
Carlo Lovadina
Keyword(s):  
Finite Element ◽  
Augmented Lagrangian ◽  
Mixed Finite Element ◽  
Lagrangian Formulations ◽  
Finite Element Schemes

scholarly journals Quasi-optimal and pressure-robust discretizations of the Stokes equations by new augmented Lagrangian formulations

2019 ◽  
Vol 40 (4) ◽  
pp. 2553-2583
Author(s):  
Christian Kreuzer ◽  
Pietro Zanotti
Keyword(s):  
Finite Element ◽  
Linear Operator ◽  
Augmented Lagrangian ◽  
Discontinuous Galerkin Methods ◽  
Stokes Equations ◽  
Galerkin Methods ◽  
Test Functions ◽  
Lagrangian Formulations ◽  
Stable Pairs ◽  
Augmented Lagrangian Formulation

Abstract We approximate the solution of the stationary Stokes equations with various conforming and nonconforming inf-sup stable pairs of finite element spaces on simplicial meshes. Based on each pair, we design a discretization that is quasi-optimal and pressure-robust, in the sense that the velocity $H^1$-error is proportional to the best velocity $H^1$-error. This shows that such a property can be achieved without using conforming and divergence-free pairs. We also bound the pressure $L^2$-error, only in terms of the best velocity $H^1$-error and the best pressure $L^2$-error. Our construction can be summarized as follows. First, a linear operator acts on discrete velocity test functions, before the application of the load functional, and maps the discrete kernel into the analytical one. Second, in order to enforce consistency, we employ a new augmented Lagrangian formulation, inspired by discontinuous Galerkin methods.

scholarly journals A UNIFIED APPROACH TO MIMETIC FINITE DIFFERENCE, HYBRID FINITE VOLUME AND MIXED FINITE VOLUME METHODS

2010 ◽  
Vol 20 (02) ◽  
pp. 265-295 ◽  
Author(s):  
JÉRÔME DRONIOU ◽  
ROBERT EYMARD ◽  
THIERRY GALLOUËT ◽  
RAPHAÈLE HERBIN
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Finite Volume ◽  
Mixed Finite Element ◽  
Finite Volume Methods ◽  
Finite Volume Scheme ◽  
Volume Scheme ◽  
Common Framework ◽  
Finite Element Schemes ◽  
Unified Method

We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the methods (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme.

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