A discontinuous Galerkin method with Lagrange multipliers for spatially-dependent advection–diffusion problems

Borker, Raunak; Farhat, Charbel; Tezaur, Radek

doi:10.1016/j.cma.2017.08.024

Cited by 12 publications

(5 citation statements)

References 23 publications

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“…As a remedy to these numerical problems, stabilization methods have been proposed in [56][57][58], and these methods are advanced [59][60][61][62] and implemented [63][64][65]. Also different numerical methods have been suggested for a robust implementation [66][67][68][69][70]. Another approach has been taken by changing the weak formulation [71][72][73][74].…”

Section: Methodsmentioning

confidence: 99%

Experimental validation of computational fluid dynamics for solving isothermal and incompressible viscous fluid flow

Abali

Savaş

2020

SN Appl. Sci.

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In order to validate a computational method for solving viscous fluid flows, experiments are carried out in an eccentric cylindrical cavity showing various flow formations over a range of Reynolds numbers. Especially, in numerical solution approaches for isothermal and incompressible flows, we search for simple experimental data for evaluating accuracy as well as performance of the computational method. Verification of different computational methods is arduous, and analytic solutions are only obtained for simple geometries like a channel flow. Clearly, a method is expected to predict different flow patterns within a cavity. Thus, we propose a configuration generating different flow formations depending on the Reynolds number and make the experimental results freely available in order to be used as an assessment criterion to demonstrate the reliability of a new computational approach.

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Section: Methodsmentioning

confidence: 99%

Experimental validation of computational fluid dynamics for solving isothermal and incompressible viscous fluid flow

Abali

Savaş

2020

SN Appl. Sci.

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“…) The solution of the system of equations (D, b) gives the pair (W, q b ), which are: Finally, on using (30) we have the vector q:…”

Section: B02 Poisson Problem With Quadratic Solutionmentioning

confidence: 99%

“…These numerical fluxes utilize additional parameters that must be evaluated. An alternative approach is adopted in the hybridized DG (HDG) method, [23][24][25][26][27] discontinuous Petrov-Galerkin (DGP) method, 28,29 or DG method with Lagrange multipliers (DGLM) 30 where the numerical fluxes are treated as additional unknowns. Among the aforementioned DG methods, the IPDG methods are the most widely used.…”

Section: Introductionmentioning

confidence: 99%

Penalty‐free discontinuous Galerkin method

Jaśkowiec,

Sukumar

2024

Numerical Meth Engineering

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In this article, we present a new high‐order discontinuous Galerkin (DG) method, in which neither a penalty parameter nor a stabilization parameter is needed. We refer to this method as penalty‐free DG. In this method, the trial and test functions belong to the broken Sobolev space, in which the functions are in general discontinuous on the mesh skeleton and do not meet the Dirichlet boundary conditions. However, a subset can be distinguished in this space, where the functions are continuous and satisfy the Dirichlet boundary conditions, and this subset is called admissible. The trial solution is chosen to lie in an augmented admissible subset, in which a small violation of the continuity condition is permitted. This subset is constructed by applying special augmented constraints to the linear combination of finite element basis functions. In this approach, all the advantages of the DG method are retained without the necessity of using stability parameters or numerical fluxes. Several benchmark problems in two dimensions (Poisson equation, linear elasticity, hyperelasticity, and biharmonic equation) on polygonal (triangles, quadrilateral, and weakly convex polygons) meshes as well as a three‐dimensional Poisson problem on hexahedral meshes are considered. Numerical results are presented that affirm the sound accuracy and optimal convergence of the method in the norm and the energy seminorm.

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“…The HDG method has been applied to various engineering problems like second order elliptic problems 21 , incompressible Navier–Stokes equations 22 , space-time fractional advection-dispersion equations 23 , or the flow in petroleum reservoirs 24 . An alternative approach is adopted in the discontinuous Petrov-Galerkin (DGP) method 25 , 26 , or the DG method with Lagrange multipliers (DGLM) 27 , where the numerical fluxes are treated as additional unknowns. Extensive overviews of some of the DG methods are presented in.douglassps2002,Kirby2005.…”

Section: Introductionmentioning

confidence: 99%

A posteriori error approximation in discontinuous Galerkin method on polygonal meshes in elliptic problems

Jaśkowiec

Pamin

2023

Sci Rep

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The paper presents a posteriori error approximation concept based on residuals in the two-dimensional discontinuous Galerkin (DG) method. The approach is relatively simple and effective in application, and it takes advantage of some unique properties of the DG method. The error function is constructed in an enriched approximation space, utilizing the hierarchical nature of the basis functions. Among many versions of the DG method, the most popular one is based on the interior penalty approach. However, in this paper a DG method with finite difference (DGFD) is utilized, where the continuity of the approximate solution is enforced by finite difference conditions applied on the mesh skeleton. In the DG methods arbitrarily shaped finite elements can be used, so in this paper the meshes with polygonal finite elements are considered, including quadrilateral and triangular elements. Some benchmark examples are presented, in which Poisson’s and linear elasticity problems are considered. The examples use various mesh densities and approximation orders to evaluate the errors. The error estimation maps, generated for the discussed tests, indicate a good correlation with the exact errors. In the last example, the error approximation concept is applied for an adaptive hp mesh refinement.

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A discontinuous Galerkin method with Lagrange multipliers for spatially-dependent advection–diffusion problems

Cited by 12 publications

References 23 publications

Experimental validation of computational fluid dynamics for solving isothermal and incompressible viscous fluid flow

Experimental validation of computational fluid dynamics for solving isothermal and incompressible viscous fluid flow

Penalty‐free discontinuous Galerkin method

A posteriori error approximation in discontinuous Galerkin method on polygonal meshes in elliptic problems

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