@article{DENG2019166,
title = "{C}onstruction of {L}ocally {C}onservative {F}luxes for {H}igh {O}rder {C}ontinuous {G}alerkin {F}inite {E}lement {M}ethods",
journal = "Journal of Computational and Applied Mathematics",
volume = "359",
pages = "166-181",
year = "2019",
issn = "0377-0427",
doi = "https://doi.org/10.1016/j.cam.2019.03.049",
url = "http://www.sciencedirect.com/science/article/pii/S0377042719301803",
author = "Quanling Deng and Victor Ginting and Bradley McCaskill",
keywords = "CGFEM, FVEM, Conservative flux, Post-processing",
abstract = "Despite their robustness, it is known that standard continuous Galerkin Finite Element Methods (CGFEMs) do not produce a locally conservative flux field. As a result, their application to solving model problems that are derived from conservation laws can be limited. To remedy this issue some form of post-processing must be performed on the CGFEM solution. In this work, a simple post-processing technique is proposed to obtain a locally conservative flux field from a CGFEM solution. One distinct advantage of the proposed method is that it produces continuous normal flux at the element’s boundary. The post-processing is implemented on nodal-centered control volumes that are constructed from the original finite element mesh. The post-processing method is performed by solving an independent set of low dimensional problems posed on each element. The associated linear algebra systems are of dimension 12(k+1)(k+2) where k is the polynomial degree of CGFEM basis on a triangular mesh. A theoretical investigation is conducted to confirm that the post-processed solution converges in an optimal fashion to the true solution in the H1 semi-norm. Various numerical examples that demonstrate the performance of technique are given. Specifically, a simulation of a model for single-phase flow in a heterogeneous system is presented to show the necessity of the local conservation as well as the effective performance of the post-processing technique."
}

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As a result, their application to solving model problems that are derived from conservation laws can be limited. To remedy this issue some form of post-processing must be performed on the CGFEM solution. In this work, a simple post-processing technique is proposed to obtain a locally conservative flux field from a CGFEM solution. One distinct advantage of the proposed method is that it produces continuous normal flux at the element’s boundary. The post-processing is implemented on nodal-centered control volumes that are constructed from the original finite element mesh. The post-processing method is performed by solving an independent set of low dimensional problems posed on each element. The associated linear algebra systems are of dimension 12(k+1)(k+2) where k is the polynomial degree of CGFEM basis on a triangular mesh. A theoretical investigation is conducted to confirm that the post-processed solution converges in an optimal fashion to the true solution in the H1 semi-norm. Various numerical examples that demonstrate the performance of technique are given. Specifically, a simulation of a model for single-phase flow in a heterogeneous system is presented to show the necessity of the local conservation as well as the effective performance of the post-processing technique.","bibtex":"@article{DENG2019166,\ntitle = \"{C}onstruction of {L}ocally {C}onservative {F}luxes for {H}igh {O}rder {C}ontinuous {G}alerkin {F}inite {E}lement {M}ethods\",\njournal = \"Journal of Computational and Applied Mathematics\",\nvolume = \"359\",\npages = \"166-181\",\nyear = \"2019\",\nissn = \"0377-0427\",\ndoi = \"https://doi.org/10.1016/j.cam.2019.03.049\",\nurl = \"http://www.sciencedirect.com/science/article/pii/S0377042719301803\",\nauthor = \"Quanling Deng and Victor Ginting and Bradley McCaskill\",\nkeywords = \"CGFEM, FVEM, Conservative flux, Post-processing\",\nabstract = \"Despite their robustness, it is known that standard continuous Galerkin Finite Element Methods (CGFEMs) do not produce a locally conservative flux field. As a result, their application to solving model problems that are derived from conservation laws can be limited. To remedy this issue some form of post-processing must be performed on the CGFEM solution. In this work, a simple post-processing technique is proposed to obtain a locally conservative flux field from a CGFEM solution. One distinct advantage of the proposed method is that it produces continuous normal flux at the element’s boundary. The post-processing is implemented on nodal-centered control volumes that are constructed from the original finite element mesh. The post-processing method is performed by solving an independent set of low dimensional problems posed on each element. The associated linear algebra systems are of dimension 12(k+1)(k+2) where k is the polynomial degree of CGFEM basis on a triangular mesh. A theoretical investigation is conducted to confirm that the post-processed solution converges in an optimal fashion to the true solution in the H1 semi-norm. Various numerical examples that demonstrate the performance of technique are given. Specifically, a simulation of a model for single-phase flow in a heterogeneous system is presented to show the necessity of the local conservation as well as the effective performance of the post-processing technique.\"\n}\n\n","author_short":["Deng, Q.","Ginting, V.","McCaskill, B."],"key":"DENG2019166","id":"DENG2019166","bibbaseid":"deng-ginting-mccaskill-constructionoflocallyconservativefluxesforhighordercontinuousgalerkinfiniteelementmethods-2019","role":"author","urls":{"Paper":"http://www.sciencedirect.com/science/article/pii/S0377042719301803"},"keyword":["CGFEM","FVEM","Conservative flux","Post-processing"],"metadata":{"authorlinks":{"ginting, v":"https://bibbase.org/show?bib=https://bibbase.org/network/files/3yQtKfRddpmAbuCJN&msg=preview&fileId=3yQtKfRddpmAbuCJN"}},"downloads":5},"search_terms":["construction","locally","conservative","fluxes","high","order","continuous","galerkin","finite","element","methods","deng","ginting","mccaskill"],"keywords":["cgfem","fvem","conservative flux","post-processing"],"authorIDs":["KGv5qNX2wproMHaMz"],"dataSources":["FMKDotGw9QMpa9Abz","x4GJj42ibi69jHYvw"]}