Two-dimensional hp adaptive finite element spaces for mixed formulations

Numerical Meth Engineering

Faria

Farias

et al. 2018

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.

Section: Finite Element Formulationsmentioning

confidence: 99%

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

Numerical Meth Engineering

Faria

Farias

et al. 2018

“…The cases of two‐dimensional triangular and rectangular geometries have already been discussed in Siqueira et al using this methodology for uniform affine partitions, and in Devloo et al, for h p ‐adaptive affine meshes. A detailed explanation is given in Castro et al for 3‐D uniform affine elements.…”

Section: Hierarchic H(div) Shape Functionsmentioning

confidence: 99%

“…As in the previous publications, the discrete mixed formulations are implemented using static condensation techniques by organizing the degrees of freedom of the flux in the form σ i ,, σ e , where σ i and σ e refer to internal and edge components of the flux, respectively. For the variable u , let u 0 be a scalar value and u i denote the remaining degrees of freedom except u 0 .…”

Section: Numerical Experimentsmentioning

confidence: 99%

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

Numerical Meth Engineering

Durán

Gomes

et al. 2017

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.

“…The vector shape functions form a hierarchy of bases, as proposed in [24] for triangular and quadrilateral affine elements, and extended to three-dimensional tetrahedral, hexahedral and prismatic affine meshes in [25]. Cases of hp-adaptive mixed formulations for other kinds of elements in two-and three-dimensional problems, including curved and surface elements, have been treated in [26,27].…”

Section: Introductionmentioning

confidence: 99%

Mixed finite element approximations of a singular elliptic problem based on some anisotropic and hp-adaptive curved quarter-point elements

Siqueira

Farias

Applied Numerical Mathematics

et al. 2020

Self Cite

Mixed finite element methods are applied to a Poisson problem with singularity at a boundary point. The approximation spaces are based on quarter-point elements, the shape functions inheriting the singular behavior of their quadratic geometric maps. Two mesh scenarios are considered, by fixing some macro quarterpoint elements at the coarse level, and subdividing them by mapping uniformly refined square meshes on the master element by their corresponding geometric transforms. For eight-noded coarse quadrilateral quarter-point elements, placing two mid-side nodes near the singular vertex, radial singularity is exactly captured along element edges, and their refinements reveal shape regular curved meshes. For an improved version, using collapsed quadrilateral quarter-point elements obtained by reducing one of the quadrilateral element edges to the singular point, the radial singularity is captured inside the coarse macro elements as well. Their uniform refinement generates anisotropic meshes, grading towards the singular point. The assembly of the required H(div)-conforming approximation spaces based on these kind of meshes are described. Results for a typical test problem demonstrate a superior effectiveness of the proposed techniques for convergence acceleration, when confronted with usual affine finite elements, for h, p and hpadaptive refinements. Specially, collapsed quarter-point elements applied to the singular problem reveal accuracy rates equivalent to standard regular contexts, of smooth solutions discretized on uniform affine meshes.