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

Castro, Douglas Azevedo; Devloo, Philippe R.B.; Farias, Agnaldo M.; Gomes, Sônia M.; Siqueira, Denise de; Durán, Omar

doi:10.1016/j.cma.2016.03.050

Cited by 29 publications

(46 citation statements)

References 21 publications

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“…Since the pioneer work by Raviart and Thomas in 1977, this formulation received considerable attention, and different kinds of approximation spaces were constructed in the 1980s . In recent years, mixed formulations have received renovated interest due to their accuracy and local conservation property …”

Section: Finite Element Formulationsmentioning

confidence: 99%

“…Approximation spaces of type

{boldP}_{k}^{* *} P_{k + 1}

: This is a space configuration recently introduced in the work of Castro et al, where the construction of approximation spaces for σ of type

{boldP}_{k}^{* *}

consists in adding to the complete vector valued spaces of type P k those interior shape functions of

{boldP}_{k + 1}^{*}

, defined by vector polynomials of degree k + 2 whose divergence is included in the scalar approximation space of type P k + 1 . Therefore, in

{boldP}_{k}^{* *}

, the face shape functions are still obtained by polynomials of degree ≤ k , but some of the internal shape functions may be obtained from polynomials of degree up to k + 2.…”

Section: Finite Element Formulationsmentioning

confidence: 99%

See 1 more Smart Citation

On continuous, discontinuous, mixed, and primal hybrid finite element methods for second‐order elliptic problems

Devloo

Faria

Farias

et al. 2018

Numerical Meth Engineering

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Summary Finite element formulations for second‐order elliptic problems, including the classic H1‐conforming Galerkin method, dual mixed methods, a discontinuous Galerkin method, and two primal hybrid methods, are implemented and numerically compared on accuracy and computational performance. Excepting the discontinuous Galerkin formulation, all the other formulations allow static condensation at the element level, aiming at reducing the size of the global system of equations. For a three‐dimensional test problem with smooth solution, the simulations are performed with h‐refinement, for hexahedral and tetrahedral meshes, and uniform polynomial degree distribution up to four. For a singular two‐dimensional problem, the results are for approximation spaces based on given sets of hp‐refined quadrilateral and triangular meshes adapted to an internal layer. The different formulations are compared in terms of L2‐convergence rates of the approximation errors for the solution and its gradient, number of degrees of freedom, both with and without static condensation. Some insights into the required computational effort for each simulation are also given.

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Section: Finite Element Formulationsmentioning

confidence: 99%

“…Approximation spaces of type

{boldP}_{k}^{* *} P_{k + 1}

: This is a space configuration recently introduced in the work of Castro et al, where the construction of approximation spaces for σ of type

{boldP}_{k}^{* *}

consists in adding to the complete vector valued spaces of type P k those interior shape functions of

{boldP}_{k + 1}^{*}

, defined by vector polynomials of degree k + 2 whose divergence is included in the scalar approximation space of type P k + 1 . Therefore, in

{boldP}_{k}^{* *}

, the face shape functions are still obtained by polynomials of degree ≤ k , but some of the internal shape functions may be obtained from polynomials of degree up to k + 2.…”

Section: Finite Element Formulationsmentioning

confidence: 99%

On continuous, discontinuous, mixed, and primal hybrid finite element methods for second‐order elliptic problems

Devloo

Faria

Farias

et al. 2018

Numerical Meth Engineering

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show abstract

“…As explained in [4], when computing sufficiently smooth solutions using Q * k Q k space configurations based on affine regular meshes, optimal convergence rates of identical approximation orders k + 1 are obtained for primal and dual variables, as well for ∇ · σ. For the Q * * k Q k+1 configuration, higher convergence rate of order k + 2 is obtained for the primal variable.…”

Section: Approximation Spacesmentioning

confidence: 99%

“…Following the developments in [4], we shall consider two stable configuration cases for approximation spaces to be used for primal u and dual σ variables in discretized versions of the mixed formulation. In both cases, the primal variable is approximated in subspaces of L 2 (Ω) formed by piecewise functions u| K = u K = Fû K , without any continuity constraint, as in typical discretized mixed formulations [2].…”

Section: Approximation Spacesmentioning

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%

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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

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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.

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Mixed finite element approximations based on 3‐Dhp‐adaptive curved meshes with two types ofH(div)‐conforming spaces

Devloo

Durán

Gomes

et al. 2017

Numerical Meth Engineering

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Two stable approximation space configurations are treated for the mixed finite element method for elliptic problems based on curved meshes. Their choices are guided by the property that, in the master element, the image of the flux space by the divergence operator coincides with the potential space. By using static condensation, the sizes of global condensed matrices, which are proportional to the dimension of border fluxes, are the same in both configurations. The meshes are composed of different topologies (tetrahedra, hexahedra, or prisms). Simulations using asymptotically affine uniform meshes, exactly fitting a spherical-like region, and constant polynomial degree distribution k, show L 2 errors of order k + 1 or k + 2 for the potential variable, while keeping order k + 1 for the flux in both configurations. The first case corresponds to RT(k) and BDFM(k+1) spaces for hexahedral and tetrahedral meshes, respectively, but holding for prismatic elements as well. The second case, further incrementing the order of approximation of the potential variable, holds for the three element topologies. The case of hp-adaptive meshes is considered for a problem modelling a porous media flow around a cylindrical horizontal well with elliptical drainage area. The effect of parallelism and static condensation in CPU time reduction is illustrated.

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

Cited by 29 publications

References 21 publications

On continuous, discontinuous, mixed, and primal hybrid finite element methods for second‐order elliptic problems

On continuous, discontinuous, mixed, and primal hybrid finite element methods for second‐order elliptic problems

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

Mixed finite element approximations based on 3‐Dhp‐adaptive curved meshes with two types ofH(div)‐conforming spaces

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