A new procedure for the construction of hierarchical high order Hdiv and Hcurl finite element spaces

Siqueira, Denise de; Devloo, Philippe R.B.; Gomes, Sônia M.

doi:10.1016/j.cam.2012.09.026

Cited by 32 publications

(25 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The fluid viscosities are µ w = 1 and µ n = 0.45. The relative permeabilities are set as in (40) with β = 2, the residual saturations are S rw = 0 and S rn = 0 (so S e = S w ) and the capillary pressure is for the non-zero capillary pressure simulation, otherwise, it is set to be zero. We ignore gravity in this test, and therefore the densities are not required.…”

Section: Example 1: Effect Of Capillary Pressurementioning

confidence: 99%

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

Arbogast

Tao

2019

Comput Geosci

View full text Add to dashboard Cite

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.

show abstract

Section: Example 1: Effect Of Capillary Pressurementioning

confidence: 99%

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

Arbogast

Tao

2019

Comput Geosci

View full text Add to dashboard Cite

show abstract

“…The numbers of these vectorial shape functions are presented in Table 1. The methodology adopted here for their construction is an extension of previous developments in [7], where the principle is to choose appropriate constant vector fields, based on the geometry of each element of a given partition of the computational region Ω, which are multiplied by an available set of H 1 hierarchical scalar basic functions to obtain the vectorial shape functions. …”

Section: Types Of Approximation Space Configurationsmentioning

confidence: 99%

“…Recently, several other papers have appeared in the literature due to increasing interest on this subject [1,7].…”

Section: Introductionmentioning

confidence: 99%

Two dimensional hierarchical mixed finite element approximations with enhanced primal variable accuracy

Farias¹,

Devloo²,

Gomes³

et al. 2017

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

Abstract. The purpose of the present paper is to analyse different possibilities of choosing balanced pairs of approximation spaces for dual (flux) and primal (pressure) variables to be used in discrete versions of the mixed finite element method for affine two dimensional meshes. In all space configurations, the principle guiding their construction is the property that the divergence of the dual space and the primal approximation space should coincide, while keeping the same order of accuracy for the flux variable and varying the accuracy order of the primal variable. There is the classic case of BDM k spaces based on triangular meshes and polynomials of total degree k for the dual variable, and k − 1 for the primal variable, showing stable simulations with optimal convergence rates of orders k + 1 and k, respectively. Another case is related to RT k and BDF M k+1 spaces for quadrilateral and triangular meshes, respectively. It gives identical approximation order k + 1 for both primal and dual variables, an improvement in accuracy obtained by increasing the degree of primal functions to k, and by enriching the dual space with some properly chosen internal shape functions of degree k + 1, while keeping degree k for the border fluxes. A new type of approximation is proposed by further incrementing the order of some internal flux functions to k + 2, and matching primal functions to k + 1 (higher than the border fluxes of degree k). Thus, higher convergence rate of order k + 2 is obtained for the primal variable. Using static condensation, the global condensed system to be solved in all the cases have same dimension (and structure), which is proportional to the space dimension of the border fluxes for each element geometry.

show abstract

“…A family of hierarchical bases BK k K = {Φ} is given, where the parameter k K refers to the total degree of the polynomials in Q k(K) used in their definitions, as proposed in [10]. The principle is to choose appropriate constant vector fieldsv, based on the geometry of the master element, which are multiplied by an available set of H 1 hierarchical scalar basic functionsφ in order to getΦ =φv.…”

Section: Approximation Spaces In H(divω)mentioning

confidence: 99%

“…Another methodology for the construction of H(div)-conforming approximation spaces is proposed in [10], which has been extended to affine hp-adaptive meshes formed by triangular or quadrilateral elements in [6], and to three-dimensional affine tetrahedral, hexahedral and prismatic meshes in [4]. The principle is to choose appropriate constant vector fields, based on the geometry of each element, which are multiplied by an available 2 set of H 1 hierarchical scalar basic functions.…”

Section: Introductionmentioning

confidence: 99%

H (div) approximations based on hp-adaptive curved meshes using quarter point elements

Siqueira¹,

Farias²,

Devloo³

et al. 2017

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

Self Cite

View full text Add to dashboard Cite

Abstract. H(div)-conforming finite element subspaces based on curved quadrilateral meshes, with hp-adaptation, are constructed to be applied in flux approximations of the mixed element formulation. In order to validate the implementation, a test problem with square-root singularity at a boundary point is simulated. The results demonstrate exponential rates of convergence, and a dramatic error reduction when quarter-point elements are applied close to the singularity. Using static condensation, the global condensed matrices to be solved have reduced dimension, which is proportional to the dimension of border fluxes.

show abstract

A new procedure for the construction of hierarchical high order Hdiv and Hcurl finite element spaces

Cited by 32 publications

References 7 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

Two dimensional hierarchical mixed finite element approximations with enhanced primal variable accuracy

H (div) approximations based on hp-adaptive curved meshes using quarter point elements

Contact Info

Product

Resources

About