Construction of 
                
                  
                
                $$H({\mathrm{div}})$$
                
                  
                    
                      H
                      (
                      div
                      )
                    
                  
                
              -conforming mixed finite elements on cuboidal hexahedra

Arbogast, Todd; Tao, Zhen

doi:10.1007/s00211-018-0998-7

Cited by 16 publications

(2 citation statements)

References 15 publications

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“…The unisolvence of the direct serendipity finite element space is proved generally in [7], or specifically for this choice of R V , R H , V and H in [6]. A nodal basis can be constructed using local linear algebra as described in [6,7].…”

Section: Direct Serendipity Elementsmentioning

confidence: 99%

A direct mixed–enriched Galerkin method on quadrilaterals for two-phase Darcy flow

Arbogast

Tao

2019

Comput Geosci

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We develop a locally conservative, finite element method for simulation of two-phase flow on quadrilateral meshes that minimize the number of degrees of freedom (DoFs) subject to accuracy requirements and the DoF continuity constraints. We use a mixed finite element method (MFEM) for the flow problem and an enriched Galerkin method (EG) for the transport, stabilized with an entropy viscosity. Standard elements for MFEM lose accuracy on quadrilaterals, so we use the newly developed AC elements which have our desired properties. Standard tensor product spaces used in EG have many excess DoFs, so we would like to use the minimal DoF serendipity elements. However, the standard elements lose accuracy on quadrilaterals, so we use the newly developed direct serendipity elements. We use the Hoteit-Firoozabadi formulation, which requires a capillary flux. We compute this in a novel way that does not break down when one of the saturations degenerate to its residual value. Extension to three dimensions is described. Numerical tests show that accurate results are obtained.

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Section: Direct Serendipity Elementsmentioning

confidence: 99%

A direct mixed–enriched Galerkin method on quadrilaterals for two-phase Darcy flow

Arbogast

Tao

2019

Comput Geosci

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“…Families of conforming finite elements defined on polygons that maintain both accuracy and a minimal number of DoFs have appeared recently [40][41][42][43] (as well as some finite elements in three dimensions [44][45][46]). The approach taken is to begin with the space of polynomials P r (E) of degree up to r defined directly on the physical element E to achieve accuracy of order r + 1.…”

Section: Introductionmentioning

confidence: 99%

Construction of Supplemental Functions for Direct Serendipity and Mixed Finite Elements on Polygons

Arbogast,

Wang

2023

Mathematics

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New families of direct serendipity and direct mixed finite elements on general planar, strictly convex polygons were recently defined by the authors. The finite elements of index r are H1 and H(div) conforming, respectively, and approximate optimally to order r+1 while using the minimal number of degrees of freedom. The shape function space consists of the full set of polynomials defined directly on the element and augmented with a space of supplemental functions. The supplemental functions were constructed as rational functions, which can be difficult to integrate accurately using numerical quadrature rules when the index is high. This can result in a loss of accuracy in certain cases. In this work, we propose alternative ways to construct the supplemental functions on the element as continuous piecewise polynomials. One approach results in supplemental functions that are in Hp for any p≥1. We prove the optimal approximation property for these new finite elements. We also perform numerical tests on them, comparing results for the original supplemental functions and the various alternatives. The new piecewise polynomial supplements can be integrated accurately, and therefore show better robustness with respect to the underlying meshes used.

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deal.II Implementation of a Two-Field Finite Element Solver for Poroelasticity

Wang

Liu

2020

Lecture Notes in Computer Science

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Construction of $$H({\mathrm{div}})$$ H ( div ) -conforming mixed finite elements on cuboidal hexahedra

Cited by 16 publications

References 15 publications

A direct mixed–enriched Galerkin method on quadrilaterals for two-phase Darcy flow

A direct mixed–enriched Galerkin method on quadrilaterals for two-phase Darcy flow

Construction of Supplemental Functions for Direct Serendipity and Mixed Finite Elements on Polygons

deal.II Implementation of a Two-Field Finite Element Solver for Poroelasticity

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