Discontinuous subgrid formulations for transport problems

Arruda, N. C. B.; Almeida, Regina C.; Carmo, Eduardo Gomes Dutra do

doi:10.1016/j.cma.2010.06.028

Cited by 7 publications

(5 citation statements)

References 23 publications

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“…To get rid of the local oscillations, many nonlinear stabilized finite element methods have been developed by adding an extra diffusivity term to the formulation to recover the monotonicity of the continuous problem [1,11,20,25,31,37,[48][49][50][51]. Such methods lead to nonlinear systems due to the introduction of nonlinear artificial dissipation dynamically adjusted in terms of the notion of scale separation.…”

Section: Introductionmentioning

confidence: 99%

A Linearized Adaptive Dynamic Diffusion Finite Element Method for Convection-Diffusion-Reaction Equations

Du,

null,

Xie

2023

AAM

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In this paper, we consider a modified nonlinear dynamic diffusion (DD) method for convection-diffusion-reaction equations. This method is free of stabilization parameters and capable of precluding spurious oscillations. We develop a reliable and efficient residual-type a posteriori error estimator, which is robust with respect to the diffusivity parameter. Furthermore, we propose a linearized adaptive DD algorithm based on the a posteriori estimator. Finally, we perform numerical experiments to verify the theoretical analysis and the performance of the adaptive algorithm.

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Section: Introductionmentioning

confidence: 99%

A Linearized Adaptive Dynamic Diffusion Finite Element Method for Convection-Diffusion-Reaction Equations

Du,

null,

Xie

2023

AAM

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“…The key feature of the method in [15] is a more flexible fine-scale basis that consists of multiple local polynomial functions that may be nonzero on element boundaries. Sub-scale models that do not vanish at element boundaries also appear elsewhere in the literature [16][17][18][19]. However, the approach in [15] is distinct in that the element boundary terms are assumed to be negligible and interelement continuity of the fine-scales is not explicitly enforced.…”

Section: Introductionmentioning

confidence: 99%

Variational Multiscale Method with Flexible Fine-Scale Basis for Diffusion-Reaction Equation: Built-In a Posteriori Error Estimate and Heterogenous Coefficients

Anguiano¹,

Masud²

2021

14th WCCM-ECCOMAS Congress

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The diffusion-reaction equation develops sharp boundary and/or internal layers for the reaction-dominated case (i.e. singularly perturbed case). In this regime, spurious oscillations pollute the solution obtained with the Galerkin finite element method (FEM). To address this issue, we employ a stabilized Variational Multiscale (VMS) method that relies on an improved expression for the fine-scale stabilization parameter that is derived via the fine-scale variational formulation facilitated by the VMS framework. The flexible fine scale basis consists of enrichment functions which may be nonzero at element edges. The stabilization parameter thus derived has spatial variation over element interiors and along inter-element boundaries that helps model the rapid variation of the residual of the Euler-Lagrange equations over the domain. This feature facilitates consistent stabilization across boundary and internal layers as well as capturing anisotropic refinement effects. In addition, VMS methods come equipped with useful a posteriori error estimators. New numerical results are presented that show the performance of this VMS method with a flexible fine-scale basis for singularly perturbed diffusion-reaction equation. These include an evaluation of the built-in error estimate for homogenous domain, and an optional modification of the method for heterogeneous domains that may result in savings in the computational cost.

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“…A great variety of Discontinuous Galerkin (DG) methods have been proposed and analyzed over the last decades for elliptic [7,15,3,4,14,16,27,8,26,5], parabolic [2,25] and hyperbolic [24,22,18,19,1,20,21] problems. Robustness, flexibility for implementing h and p-adaptivity strategies and easy parallelization are well known advantages of DG methods arising from the use of broken finite element spaces.…”

Section: Introductionmentioning

confidence: 99%

“…Based on the ideas supporting hybridization techniques and the MDG method we propose in [5] the locally discontinuous but globally continuous finite element method (LDGC). This method combines the advantages of Discontinuous Galerkin methods with the element based data structure and reduced computational cost of classical conforming finite element methods.…”

Section: Introductionmentioning

confidence: 99%

“…The proposed method can be viewed as a stabilized hybrid formulation consisting of locally discontinuous Galerkin problems coupled to a globally continuous problem. In [5] the LDGC method is presented and analyzed for Poisson problem, according to DG framework studied in [4]. In this paper we present the LDGC method for Reaction-Diffusion problems, and the main results of the numerical analysis, following the approach presented in [25].…”

Section: Introductionmentioning

confidence: 99%

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Numerical Analysis of a Locally Projected Discontinuous Galerkin Method for Elliptic Problems

Arruda¹,

Loula²,

Almeida³

2014

Proceedings of 10th World Congress on Computational Mechanics

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In this paper we study a new discontinuous Galerkin method which uses a computational structure compatible with conforming finite element methods, reducing considerably the number of degrees-of-freedom. The Locally Discontinuous but Globally Continuous Galerkin method starts with a discontinuous finite element space and constructs a continuous representation for it by means of local projections. The used technique is similar to that employed in hybridizable methods, and discontinuous solution is recovered by solving local element-wise problems. We present the numerical analysis of the method and numerical results to confirm the predicted convergence rates. Moreover, numerical experiments are conducted in order to evaluate the better performance of this formulation when compared to the continuous or discontinuous Galerkin formulations.

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Discontinuous subgrid formulations for transport problems

Cited by 7 publications

References 23 publications

A Linearized Adaptive Dynamic Diffusion Finite Element Method for Convection-Diffusion-Reaction Equations

A Linearized Adaptive Dynamic Diffusion Finite Element Method for Convection-Diffusion-Reaction Equations

Variational Multiscale Method with Flexible Fine-Scale Basis for Diffusion-Reaction Equation: Built-In a Posteriori Error Estimate and Heterogenous Coefficients

Numerical Analysis of a Locally Projected Discontinuous Galerkin Method for Elliptic Problems

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