Denise de Siqueira

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

This paper considers a systematic procedure for the construction of a hierarchy of high order finite element approximation for Hdiv and Hcurl spaces based on triangular and quadrilateral partitions of bidimensional domains. The principle is to choose an appropriate set of vectors, based on the geometry of each element, which are multiplied by an available set of H^1 hierarchical scalar basic functions. This strategy produces vector basis functions with continuous normal or tangent components on the elements interfaces, properties that characterise functions in Hdiv or Hcurl, respectively. We also present a numerical study to evaluate the correct balancedness of the resulting Hdiv spaces of degree k and L^2 spaces of degree k-1 on the resolution of the mixed formulation for a Steklov eigenvalue problem.

Two-dimensional hp adaptive finite element spaces for mixed formulations

One important characteristic of mixed finite element methods is their ability to provide accurate and locally conservative fluxes, an advantage over standard H 1 -finite element discretizations. However, the development of p or h p adaptive strategies for mixed formulations presents a challenge in terms of code complexity for the construction of H ( d i v ) -conforming shape functions of high order on non-conforming meshes, and compatibility verification of the approximation spaces for primal and dual variables ( i n f - s u p condition). In the present paper, a methodology is presented for the assembly of such approximation spaces based on quadrilateral and triangular meshes. In order to validate the computational implementations, and to show their consistent applications to mixed formulations, elliptic model problems are simulated to show optimal convergence rates for h and p refinements, using uniform and non-uniform (non-conformal) settings for a problem with smooth solution, and using adaptive h p -meshes for the approximation of a solution with strong gradients. Results for similar simulations using H 1 -conforming formulation are also presented, and both methods are compared in terms of accuracy and required number of degrees of freedom using static condensation.

Error analysis for a new mixed finite element method in 3D

There are 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 elliptic problems arising in fluid simulations. Three cases shall be studied for discretized three dimensional formulations, based on tetrahedral, hexahedral, and prismatic meshes. The principle guiding the constructions of the approximation spaces 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. Some cases correspond either to the classic spaces of Raviart-Thomas, Brezzi-Douglas-Marini, Brezzi-Douglas-Fortin-Marini or Nedelec types. A new kind of approximation is proposed by further incrementing the order of some internal flux functions, and matching primal functions at the border fluxes. In this article we develop a unified error analysis for all these space families, and element geometries.

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

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 BDMk 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 RTk and BDF Mk+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.

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

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.

Two dimensional mixed finite element approximations for elliptic problems with enhanced accuracy for the potential and flux divergence

Abstract The purpose of the present paper is to analyse two new different possibilities of choosing balanced pairs of approximation spaces for dual (flux) and primal (potential) variables, one for triangles and the other one for quadrilateral elements, to be used in discrete versions of the mixed finite element method for elliptic problems. They can be interpreted as enriched versions of B D F M k + 1 spaces based on triangles, and of R T k spaces for quadrilateral elements. The new flux approximations are incremented with properly chosen internal shape functions (with vanishing normal components on the edges) of degree k + 2 , and matching primal functions of degree k + 1 (higher than the border fluxes, which are kept of degree k ). In all these cases, the divergence of the flux space coincide with the primal approximation space on the master element, producing stable simulations. Using static condensation, the global condensed system to be solved in the enriched cases has same dimension (and structure) of the original ones, which is proportional to the space dimension of the border fluxes for each element geometry. Measuring the errors with L 2 -norms, the enriched space configurations give higher convergence rate of order k + 2 for the primal variable, while keeping the order k + 1 for the flux. For affine meshes, the divergence errors have the same improved accuracy rate as for the error in the primal variable. For quadrilateral non-affine meshes, for instance trapezoidal elements, the divergence error has order k + 1 , one unit more than the order k occurring for R T k spaces on this kind of deformed meshes. This fact also holds for A B F k elements, but for them the potential order of accuracy does not improve, keeping order k + 1 .

Two-Dimensional H ( div )-Conforming Finite Element Spaces with hp -Adaptivity

The purpose of the paper is to analyse the effect of hp mesh adaptation when discretized versions of finite element mixed formulations are applied to elliptic problems with singular solutions. Two stable configurations of approximation spaces, based on affine triangular and quadrilateral meshes, are considered for primal and dual (flux) variables. When computing sufficiently smooth solutions using regular meshes, the first configuration gives optimal convergence rates of identical approximation orders for both variables, as well as for the divergence of the flux. For the second configuration, higher convergence rates are obtained for the primal variable. Furthermore, after static condensation is applied, the condensed systems to be solved have the same dimension in both configuration cases, which is proportional to their border flux dimensions. A test problem with a steep interior layer is simulated, and the results demonstrate exponential rates of convergence. Comparison of the results obtained with H1-conforming formulation are also presented.

Efeitos de refinamentos h e p para subespacos de funcoes do tipo Hdiv em malhas irregulares

E sabido que o emprego de formulacoes mistas e mais apropriado para simulacoes de fluxos em meios nao homogeneos do que o modelo classico em termos apenas da pressao. Tais formulacoes tem como caracteristica a aproximacao simultânea dos campos de pressao e velocidade, podendo fornecer boas solucoes numericas para esses campos ao mesmo tempo em que garante a conservacao local de massa. No entanto para adotar tal estrategia e necessaria a construcao de bases para subespacos de elementos finitos do tipo Hdiv para o fluxo que sejam balanceados com subespacos de elementos finitos para a pressao. Em [2] foi apresentada uma sistematica de construcao de bases para subespacos de Hdiv usando funcoes vetoriais polinomiais de grau uniforme em malhas triangulares e quadrilaterais. O presente trabalho tem como objetivo apresentar o desempenho das aproximacoes construidas na presenca de refinamentos h e p em malhas irregulares.

Hierarchical High Order Finite Element Approximation Spaces for H(div) and H(curl)

The aim of this paper is to present a systematic procedure for the construction of a hierarchy of high order finite element approximations for H(div) and H(curl) spaces based on quadrilateral and triangular elements with rectilinear edges. The principle is to chose appropriate vector fields, based on the geometry of each element, which are multiplied by an available set of H 1 hierarchical scalar basic functions. We show that the resulting local vector bases can be combined to obtain continuous normal or tangent components on the elements interfaces, properties that characterize piecewise polynomial functions in H(div) or H(curl), respectively.