An immersed weak Galerkin method for elliptic interface problems

Mu, Lin; Zhang, Xu

doi:10.1016/j.cam.2018.08.023

Cited by 13 publications

(2 citation statements)

References 40 publications

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“…Concerning the discretization methods for conjugate heat transfer problems (often referred to as elliptic interface problems), a comprehensive literature can be found including finite difference methods, 21,22 finite element methods, [23][24][25] discontinuous Galerkin methods, 26 finite volume methods, [27][28][29][30] lattice Boltzmann methods, 31,32 among others. Finite volume methods are particularly interesting in the context of heat and mass transfer problems, given the conservation property intrinsically preserved and the ease of handling general unstructured meshes.…”

Section: Conjugate Heat Transfer Modelingmentioning

confidence: 99%

High‐order accurate conjugate heat transfer solutions with a finite volume method in anisotropic meshes with application in polymer processing

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Clain

et al. 2021

Numerical Meth Engineering

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The computational modeling has become an indispensable tool to support the engineering design and control of many industrial processes. In polymer processing applications, the simulation of conjugate heat transfer phenomena is critical as rigorous temperature control is necessary to ensure that the produced parts meet the required specifications. In this article, a high-order accurate finite volume method is proposed to improve the numerical accuracy and the computational efficiency of conjugate heat transfer simulations with application in polymer processing. Anisotropic meshes are investigated to significantly reduce the number of unknowns in convection-dominated problems where the higher temperature variations occur perpendicularly to curved boundaries and interfaces. The reconstruction for off-site data method based on polynomial reconstructions is employed to fulfill the prescribed boundary and interface conditions solely using polygonal meshes to avoid the limitations of curved mesh approaches. A code verification benchmark based on manufactured solutions proves that the proposed method provides a fourth-order of convergence and is computationally more cost-effective than the classical second-order of convergence. Moreover, meshes with higher aspect ratios improve the calculation efficiency but suffer from a small accuracy penalty due to conditioning deterioration. Comparison with the classical finite volume discretization techniques, widely implemented in commercial and open-source software packages, is also provided. The applicability and performance of the proposed method are further supported with a practical case study for the sheet extrusion cooling stage. K E Y W O R D Sanisotropic polygonal meshes, conjugate heat transfer problems, curved boundaries and interfaces, high-order accurate finite volume method, OpenFOAM® software, polymer processing applications 1146

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Section: Conjugate Heat Transfer Modelingmentioning

confidence: 99%

High‐order accurate conjugate heat transfer solutions with a finite volume method in anisotropic meshes with application in polymer processing

Costa

Nóbrega

Clain

et al. 2021

Numerical Meth Engineering

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“…Furthermore, it is known that WG schemes are absolutely stable, and the corresponding solutions generally preserve the physical quantities inherited by the modeling equations at discrete levels. WG-FEM has been developed for many PDEs including the linear elasticity equation [33], the Stokes equation [44], Maxwell's equation [22,29], the elliptic interface problem [23], the Brinkman equation [17], the Helmholtz equation [24], the Sobolev equation [8], and the wave equation [11] etc. The latest development of WG-FEM is the primal-dual weak Galerkin finite element method for second order elliptic equations in non-divergence form [31] and the Fokker-Planck equation [32].…”

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confidence: 99%

Superconvergence of the gradient approximation for weak Galerkin finite element methods on nonuniform rectangular partitions

Wang

2020

Applied Numerical Mathematics

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This article presents a superconvergence for the gradient approximation of the second order elliptic equation discretized by the weak Galerkin finite element methods on nonuniform rectangular partitions. The result shows a convergence of O(h r ), 1.5 ≤ r ≤ 2, for the numerical gradient obtained from the lowest order weak Galerkin element consisting of piecewise linear and constant functions. For this numerical scheme, the optimal order of error estimate is O(h) for the gradient approximation. The superconvergence reveals a superior performance of the weak Galerkin finite element methods. Some computational results are included to numerically validate the superconvergence theory.Key words. weak Galerkin, finite element methods, second order elliptic equations, superconvergence, nonuniform rectangular partitions.

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