A uniformly stable family of mixed hp-finite elements with continuous pressures for incompressible flow

A new family of mixed hp-finite elements is presented for thediscretization of planar Stokes flow on meshes of curvilinear,quadrilateral elements. The elements involve continuous pressuresand are shown to be stable with an inf–sup constant boundedbelow independently of the mesh-size h and the spectral orderp. The spaces have balanced approximation properties—theorders of approximation in h and p are equal for both the velocityand the pressure. This is the first example of a uniformly stablemethod with continuous pressures for spectral element discretizationof Stokes equations, valid for geometrically refined meshesand curvilinear elements.     

The hp-MITC finite element method for the Reissner–Mindlin plate problem

The popular MITC finite elements used for the approximation of the Reissner–Mindlin plate are extended to the case where elements of non-uniform degree p distribution are used on locally refined meshes. Such an extension is of particular interest to the hp-version and hp-adaptive finite element methods. A priori error bounds are provided showing that the method is locking-free. The analysis is based on new approximation theoretic results for non-uniform Brezzi–Douglas–Fortin–Marini spaces, and extends the results obtained in the case of uniform order approximation on globally quasi-uniform meshes presented by Stenberg and Suri (SIAM J. Numer. Anal. 34 (1997) 544). Numerical examples illustrating the theoretical results and comparing the performance with alternative standard Galerkin approaches are presented for two new benchmark problems with known analytic solution, including the case where the shear stress exhibits a boundary layer. The new method is observed to be locking-free and able to provide exponential rates of convergence even in the presence of boundary layers.     

Exponential convergence of mixed hp-DGFEM for Stokes flow in polygons

Summary. We analyze mixed hp-discontinuous Galerkin finite element methods (DGFEM) for Stokes flow in polygonal domains. In conjunction with geometrically refined quadrilateral meshes and linearly increasing approximation orders, we prove that the hp-DGFEM leads to exponential rates of convergence for piecewise analytic solutions exhibiting singularities near corners. Mathematics Subject Classification (2000):65N30     

Mixed hp-DGFEM for incompressible flows II: Geometric edge meshes

We consider the Stokes problem of incompressible fluid flowin three-dimensional polyhedral domains discretized on hexahedralmeshes with hp-discontinuous Galerkin finite elements of typeQ_k for the velocity and Q_k–1 for the pressure. We provethat these elements are inf-sup stable on geometric edge meshesthat are refined anisotropically and non-quasiuniformly towardsedges and corners. The discrete inf-sup constant is shown tobe independent of the aspect ratio of the anisotropic elementsand is of O(k^–3/2) in the polynomial degree k, as in thecase of conforming Q_k–Q_k–2 approximations on thesame meshes.     

A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations

The aim of this paper is to develop an hp-version a posteriori error analysis for the time discretization of parabolic problems by the continuous Galerkin (cG) and the discontinuous Galerkin (dG) time-stepping methods, respectively. The resulting error estimators are fully explicit with respect to the local time-steps and approximation orders. Their performance within an hp-adaptive refinement procedure is illustrated with a series of numerical experiments.     

Domain decomposition preconditioners of Neumann-Neumann type for hp-approximations on boundary layer meshes in three dimensions

We develop and analyse Neumann–Neumann methods for hpfinite-element approximations of scalar elliptic problems ongeometrically refined boundary layer meshes in three dimensions.These are meshes that are highly anisotropic where the aspectratio typically grows exponentially with the polynomial degree.The condition number of our preconditioners is shown to be independentof the aspect ratio of the mesh and of potentially large jumpsof the coefficients. In addition, it only grows polylogarithmicallywith the polynomial degree, as in the case of p approximationson shape-regular meshes. This work generalizes our previousone on two-dimensional problems in Toselli & Vasseur (2003a,submitted to Numerische Mathematik, 2003c to appear in Comput.Methods Appl. Mech. Engng.) and the estimates derived here canbe employed to prove condition number bounds for certain typesof FETI methods.     

A hp-discontinuous Galerkin method for the time-dependent Maxwell’s equation: a priori error estimate

A discontinuous Galerkin method for the numerical approximation for the time-dependent Maxwell’s equations in “stable medium” with supraconductive boundary, is introduced and analysed. its hp-analysis is carried out and error estimates that are optimal in the meshsize h and slightly suboptimal in the approximation degree p are obtained.     

Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method

We consider the hp-version interior penalty discontinuous Galerkinfinite-element method (hp-DGFEM) for second-order linear reaction–diffusionequations. To the best of our knowledge, the sharpest knownerror bounds for the hp-DGFEM are due to Rivière et al.(1999,Comput. Geosci., 3, 337–360) and Houston et al.(2002,SIAM J. Numer. Anal., 99, 2133–2163). These are optimalwith respect to the meshsize h but suboptimal with respect tothe polynomial degree p by half an order of p. We present improvederror bounds in the energy norm, by introducing a new functionspace framework. More specifically, assuming that the solutionsbelong element-wise to an augmented Sobolev space, we deducefully hp-optimal error bounds.     

Locking-free DGFEM for elasticity problems in polygons

The h-version of the discontinuous Galerkin finite element method(h-DGFEM) for nearly incompressible linear elasticity problemsin polygons is analysed. It is proved that the scheme is robust(locking-free) with respect to volume locking, even in the absenceof H²-regularity of the solution. Furthermore, it is shown thatan appropriate choice of the finite element meshes leads torobust and optimal algebraic convergence rates of the DGFEMeven if the exact solutions do not belong to H².     

Analysis of a semidiscrete discontinuous Galerkin scheme for compressible miscible displacement problem

Numerical approximation of the coupled system of compressible miscible displacement problem in porous media is considered in this paper. A continuous in time discontinuous Galerkin scheme is developed. The symmetric interior penalty discontinuous Galerkin method is used to solve both the flow and transport equations. Upwind technique is used to treat the convection term in the transport equation. The hp-a priori error bounds are derived.     

Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows

This paper is concerned with the numerical simulation of time-dependent variably-saturated Darcian flow problems described by the Richards equation. We present the adaptive higher-order space-time discontinuous Galerkin (hp-STDG) method which optimizes accuracy and efficiency by balancing the errors that arise from the space and time discretizations and from the resulting nonlinear algebraic system. Convergence problems related to the transition between unsaturated flow and saturated flow are eliminated by regularizing the constitutive formulas. We also present an hp-anisotropic mesh adaptation technique capable of generating unstructured triangular elements with optimal sizes, shapes, and polynomial approximation degrees. Several numerical experiments are presented to demonstrate the accuracy, efficiency, and robustness of the numerical method presented here.     

The combination technique for a two-dimensional convection-diffusion problem with exponential layers

Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method with Shishkin meshes. Writing N for the maximum number of mesh intervals in each coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin FEM on and meshes. It is shown that the combination FEM yields (up to a factor ln N) the same order of accuracy in the associated energy norm as the Galerkin FEM on an N × N mesh, but it requires only (N ^3/2) degrees of freedom compared with the (N ²) used by the Galerkin FEM. An analogous result is also proved for the streamline diffusion finite element method. This work was supported by the National Natural Science Foundation of China (10701083 and 10425105), the Chinese National Basic Research Program (2005CB321704) and the Boole Centre for Research in Informatics at National University of Ireland Cork.     

Mixed HP-finite element approximations on geometric edge and boundary layer meshes in three dimensions

Summary. In this paper, we consider the Stokes problem in a three-dimensional polyhedral domain discretized with hp finite elements of type ?^k for the velocity and ?^k?2 for the pressure, defined on hexahedral meshes anisotropically and non quasi-uniformly refined towards faces, edges, and corners. The inf-sup constant of the discretized problem is independent of arbitrarily large aspect ratios. Our work generalizes a recent result for two-dimensional problems in [10, 11].     

A weak Galerkin mixed finite element method for the Helmholtz equation with large wave numbers

In this article, a new weak Galerkin mixed finite element method is introduced and analyzed for the Helmholtz equation with large wave numbers. The stability and well‐posedness of the method are established for any wave number k without mesh size constraint. Allowing the use of discontinuous approximating functions makes weak Galerkin mixed method highly flexible in term of little restrictions on approximations and meshes. In the weak Galerkin mixed finite element formulation, approximation functions can be piecewise polynomials with different degrees on different elements and meshes can consist elements with different shapes. Suboptimal order error estimates in both discrete H¹ and L² norms are established for the weak Galerkin mixed finite element solutions. Numerical examples are tested to support the theory.     

hp–Adaptive composite discontinuous Galerkin methods for elliptic problems on complicated domains

In this article, we develop the a posteriori error estimation of hp–version discontinuous Galerkin composite finite element methods for the discretization of second‐order elliptic partial differential equations. This class of methods allows for the approximation of problems posed on computational domains which may contain a huge number of local geometrical features, or microstructures. Although standard numerical methods can be devised for such problems, the computational effort may be extremely high, as the minimal number of elements needed to represent the underlying domain can be very large. In contrast, the minimal dimension of the underlying composite finite element space is independent of the number of geometric features. Computable bounds on the error measured in terms of a natural (mesh‐dependent) energy norm are derived. Numerical experiments highlighting the practical application of the proposed estimators within an automatic hp–adaptive refinement procedure will be presented. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1342–1367, 2014     

A note on the design of hp-version interior penalty discontinuous Galerkin finite element methods for degenerate problems

^** Email: emmanuil.georgoulis{at}mcs.le.ac.uk^*** Email: al{at}maths.strath.ac.uk We consider a variant of the hp-version interior penalty discontinuousGalerkin finite element method (IP-DGFEM) for second-order problemsof degenerate type. We do not assume uniform ellipticity ofthe diffusion tensor. Moreover, diffusion tensors of arbitraryform are covered in the theory presented. A new, refined recipefor the choice of the discontinuity-penalization parameter (thatis present in the formulation of the IP-DGFEM) is given. Makinguse of the recently introduced augmented Sobolev space framework,we prove an hp-optimal error bound in the energy norm and anh-optimal and slightly p-suboptimal (by only half an order ofp) bound in the L² norm (the latter, for the symmetric versionof the IP-DGFEM), provided that the solution belongs to an augmentedSobolev space.     

Error Analysis of Galerkin Methods for Dirichlet Problems Containing Boundary Singularities

This paper deals with a two-dimensional Poisson problem definedin a polygonal domain and having inhomogeneous Dirichlet boundaryconditions. A Galerking method for obtaining approximationsusing triangular finite elements is described. The situationwhen the exact solution contains a singularity at a re-entrantcorner of the boundary is considered, and for this an O(h) boundon the error in the Galerkin approximation is derived.     

A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges

The solution of the Stokes problem in three-dimensional domainswith edges has anisotropic singular behaviour which is treatednumerically by using anisotropic finite element meshes. Thevelocity is approximated by Crouzeix–Raviart (nonconformingP₁ ) elements and the pressure by piecewise constants. Thismethod is stable for general meshes (without minimal or maximalangle condition). Denoting by N_e the number of elements in themesh, the interpolation and consistency errors are of the optimalorder h N_e^–1/3 which is proved for tensor product meshes.As a by-product, we analyse also nonconforming prismatic elementswith P₁ [oplus ] span {x₃²} as the local space for the velocitywhere x₃ is the direction of the edge.     

A Finite-element Method for Solving Elliptic Equations with Neumann Data on a Curved Boundary Using Unfitted Meshes

This paper considers a finite-element approximation of a Poissonequation in a region with a curved boundary on which a Neumanncondition is prescribed. Piecewise linear and bilinear elementsare used on unfitted meshes with the region of integration beingreplaced by a polygonal approximation. It is shown, despitethe variational crimes, that the rate of convergence is stillorder (h) in the H¹ norm. Numerical examples show that the methodis easy to implement and that the predicted rate of convergenceis obtained. Supported by SERC postdoctoral fellowship RF/5830.     

A local error analysis of the boundary-concentrated hp-FEM

^** Email: teibner{at}mathematik.tu-chemnitz.de^*** Email: melenk{at}tuwien.ac.at The boundary-concentrated finite-element method (FEM) is a variantof the hp-version of the FEM that is particularly suited forthe numerical treatment of elliptic boundary value problemswith smooth coefficients and boundary conditions with low regularityor non-smooth geometries. In this paper, we consider the caseof the discretization of a Dirichlet problem with the exactsolution u H¹⁺() and investigate the local error in variousnorms. For 2D problems, we show that the error measured in thesenorms is O(N^––ß), where N denotes thedimension of the underlying finite-element space and ß> 0. Furthermore, we present a new Gauss–Lobatto-basedinterpolation operator that is adapted to the case of non-uniformpolynomial degree distributions.