Agnaldo M. Farias

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.

Hierarchical high order finite element bases for H ( div ) spaces based on curved meshes for two-dimensional regions or manifolds

The mixed finite element formulation for elliptic problems is characterized by simultaneous calculations of the potential (primal variable) and of the flux field (dual variable). This work focuses on new H ( div ) -conforming finite element spaces, which are suitable for flux approximations, based on curved meshes of a planar region or a manifold domain embedded in R 3 . The adopted methodology for the construction of H ( div ) bases consists in using hierarchical H 1 -conforming scalar bases multiplied by vector fields that are properly constructed on the master element and mapped to the geometrical elements by the Piola transformation, followed by a normalization procedure. They are classified as being of edge or internal type. The normal component of an edge function coincides on the corresponding edge with the associated scalar shape function, and vanishes over the other edges, and the normal components of an internal shape function vanishes on all element edges. These properties are fundamental for the global assembly of H ( div ) -conforming functions locally defined by these vectorial shape functions. For applications to the mixed formulation, the configuration of the approximation spaces is such that the divergence of the dual space and the primal approximation space coincides. Results of verification numerical tests are presented for curved triangular and quadrilateral partitions on circular, cylindrical and spherical regions, demonstrating stable convergence with optimal convergence rates, coinciding for primal and dual variables. Innovative two dimensional Hdiv approximation spaces for curved quadrilateral and triangular elements.Hdiv bases properly constructed on the master element and mapped to the geometrical elements by the Piola transformation, followed by a normalization procedure.Stable convergence with optimal convergence rates, coinciding for primal and dual variables.Study of number of condensed equations as a function of the polynomial order of the shape functions.

Application of a combined continuous–discontinuous Galerkin finite element method for the solution of the Girkmann problem

Abstract In January 2008, the International Association of Computational Mechanics (IACM) invited the scientific community to solve the Girkmann problem by finite element methods. The challenge was to get certain quantities of interest in terms of the approximate solutions within 5% of accuracy of their exact counterparts. For instance, the goal could be to evaluate the bending moment or the shear force at the junction of the shell and the ring. The purpose of the present paper is to solve an axisymmetric solid elastic model for this problem using a continuous finite element method in combination with discontinuous interface elements in the region of interest (named as the DG-FEM method). The numerical results of the p version of the DG-FEM method are presented and discussed. The results are verified with respect to previous results published in the literature showing excellent consistent results.

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.