Implementation of finite element nonlinear Galerkin methods using hierarchical bases

1993 ◽  
Vol 11 (5-6) ◽  
pp. 384-407 ◽  
Author(s):  
Jacques Laminie ◽  
Fr�d�ric Pascal ◽  
Roger Temam
Keyword(s):  
Finite Element ◽  
Galerkin Methods ◽  
Hierarchical Bases ◽  
Nonlinear Galerkin Methods

scholarly journals Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems

2020 ◽  
Vol 85 (2) ◽  
Author(s):  
R. Abgrall ◽  
J. Nordström ◽  
P. Öffner ◽  
S. Tokareva
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Discontinuous Galerkin ◽  
Finite Element Methods ◽  
Main Idea ◽  
Galerkin Methods ◽  
Exact Integration ◽  
Galerkin Scheme ◽  
Continuous Galerkin ◽  
Continuous Galerkin Methods

AbstractIn the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.

Nonlinear Galerkin methods for a system of PDEs with Turing instabilities

CALCOLO ◽  
2018 ◽  
Vol 55 (1) ◽  
Author(s):  
Konstantinos Spiliotis ◽  
Lucia Russo ◽  
Francesco Giannino ◽  
Salvatore Cuomo ◽  
Constantinos Siettos ◽  
...  
Keyword(s):  
Galerkin Methods ◽  
Turing Instabilities ◽  
Nonlinear Galerkin Methods

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.

Nonlinear Galerkin Methods

1989 ◽  
Vol 26 (5) ◽  
pp. 1139-1157 ◽  
Author(s):  
Martine Marion ◽  
Roger Temam
Keyword(s):  
Galerkin Methods ◽  
Nonlinear Galerkin Methods
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