Recent Advances in Least-Squares and Discontinuous Petrov–Galerkin Finite Element Methods

Bertrand, Fleurianne; Demkowicz, Leszek; Gopalakrishnan, Jay; Heuer, Norbert

doi:10.1515/cmam-2019-0097

Cited by 13 publications

(7 citation statements)

References 18 publications

(17 reference statements)

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“…where Ω is a bounded domain in R 3 with Lipschitz boundary Γ, (0,T] a time interval, x= (x,y,z) ⊤ the position variable, and ∇ = ∂ ∂x , ∂ ∂y , ∂ ∂z ⊤ the gradient vector. In (2.1), v(x,t) = (v 1 (x,t),v 2 (x,t),v 3 (x,t)) ⊤ is the velocity field and u 0 (x) is the initial condition.…”

Section: Characteristic Finite Element Methodsmentioning

confidence: 99%

“…In case of convection-dominated flows, the conventional Eulerian finite element methods use up-stream weighting in their implementations to stabilize the discretization. For example, the most popular Eulerian finite element methods are the streamline upwind Petrov-Galerkin methods [3,5,11], the Taylor-Galerkin methods [8,12,16] and the Galerkin/least-squares methods [3,9,26]. However, truncation errors generated by the time integration in these conventional Eulerian methods are dominant and require Courant-Friedrichs-Lewy (CFL) stability conditions which impose sever restrictions on the time steps used in the numerical computations.…”

Section: Introductionmentioning

confidence: 99%

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A Conservative and Monotone Characteristic Finite Element Solver for Three-Dimensional Transport and Incompressible Navier-Stokes Equations on Unstructured Grids

Khouya¹

2022

CICP

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We propose a mass-conservative and monotonicity-preserving characteristic finite element method for solving three-dimensional transport and incompressible Navier-Stokes equations on unstructured grids. The main idea in the proposed algorithm consists of combining a mass-conservative and monotonicity-preserving modified method of characteristics for the time integration with a mixed finite element method for the space discretization. This class of computational solvers benefits from the geometrical flexibility of the finite elements and the strong stability of the modified method of characteristics to accurately solve convection-dominated flows using time steps larger than its Eulerian counterparts. In the current study, we implement three-dimensional limiters to convert the proposed solver to a fully mass-conservative and essentially monotonicity-preserving method in addition of a low computational cost. The key idea lies on using quadratic and linear basis functions of the mesh element where the departure point is localized in the interpolation procedures. The proposed method is applied to well-established problems for transport and incompressible Navier-Stokes equations in three space dimensions. The numerical results illustrate the performance of the proposed solver and support its ability to yield accurate and efficient numerical solutions for three-dimensional convection-dominated flow problems on unstructured tetrahedral meshes.

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Section: Characteristic Finite Element Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Conservative and Monotone Characteristic Finite Element Solver for Three-Dimensional Transport and Incompressible Navier-Stokes Equations on Unstructured Grids

Khouya¹

2022

CICP

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“…Most of these computational techniques can be classified into three main categories: (i) Eulerian methods, (ii) Lagrangian methods and (iii) semi-Lagrangian methods. In the framework of finite elements, the most popular Eulerian methods are the streamline upwind Petrov-Galerkin methods [9,11], Galerkin/least-squares methods [9,26] and Taylor-Galerkin methods [12,13]. However, it is well known that these Eulerian methods do not perform very satisfactory in the case of convection-dominated problems unless small time steps and highly refined grids are used in the simulations.…”

Section: Introductionmentioning

confidence: 99%

Bernstein-Bézier Galerkin-Characteristics Finite Element Method for Convection-Diffusion Problems

El‐Amrani

El-Kacimi

Khouya³

et al. 2022

J Sci Comput

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A class of Bernstein-Bézier basis based high-order finite element methods is developed for the Galerkin-characteristics solution of convection-diffusion problems. The Galerkin-characteristics formulation is derived using a semi-Lagrangian discretization of the total derivative in the considered problems. The spatial discretization is performed using the finite element method on unstructured meshes. The Lagrangian interpretation in this approach greatly reduces the time truncation errors in the Eulerian methods. To achieve high-order accuracy in the Galerkin-characteristics solver, the semi-Lagrangian method requires high-order interpolating procedures. In the present work, this step is carried out using the Bernstein-Bézier basis functions to evaluate the solution at the departure points. Triangular Bernstein-Bézier patches are constructed in a simple and inherent manner over finite elements along the characteristics. An efficient preconditioned conjugate gradient solver is used for the linear systems of algebraic equations. Several numerical examples including advection-diffusion equations with known analytical solutions and the viscous Burgers problem are considered to illustrate the accuracy, robustness and performance of the proposed approach. The computed results support our expectations for a stable and highly accurate Bernstein-Bézier Galerkin-characteristics finite element method for convection-diffusion problems.

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“…There is a vast amount of literature on various techniques designed to take care of the numerical instabilities associated with this type of equations. We refer the reader to recent and classical works on the subject focused on some of these techniques: mixed FE methods [11,8,19,7,10]; discontinuous Galerkin methods [12,15,17,20]; discontinuous Petrov-Galerkin methods [5,14,13].…”

Section: Introductionmentioning

confidence: 99%

On the unisolvence for the quasi-polynomial spaces of differential forms

Wu,

Zikatanov

2020

Preprint

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We consider quasi-polynomial spaces of differential forms defined as weighted (with a positive weight) spaces of differential forms with polynomial coefficients. We show that the unisolvent set of functionals for such spaces on a simplex in any spatial dimension is the same as the set of such functionals used for the polynomial spaces. The analysis in the quasi-polynomial spaces, however, is not standard and requires a novel approach. We are able to prove our results without the use of Stokes' Theorem, which is the standard tool in showing the unisolvence of functionals in polynomial spaces of differential forms. These new results provide tools for studying exponentially-fitted discretizations stable for general convection-diffusion problems in Hilbert differential complexes.

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Recent Advances in Least-Squares and Discontinuous Petrov–Galerkin Finite Element Methods

Cited by 13 publications

References 18 publications

A Conservative and Monotone Characteristic Finite Element Solver for Three-Dimensional Transport and Incompressible Navier-Stokes Equations on Unstructured Grids

A Conservative and Monotone Characteristic Finite Element Solver for Three-Dimensional Transport and Incompressible Navier-Stokes Equations on Unstructured Grids

Bernstein-Bézier Galerkin-Characteristics Finite Element Method for Convection-Diffusion Problems

On the unisolvence for the quasi-polynomial spaces of differential forms

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