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     

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.     

A posteriori error estimates of hp spectral element methods for optimal control problems with L2-norm state constraint

In this paper, we investigate a distributed optimal control problem governed by elliptic partial differential equations with L²-norm constraint on the state variable. Firstly, the control problem is approximated by hp spectral element methods, which combines the advantages of the finite element methods with spectral methods; then, the optimality conditions of continuous system and discrete system are presented, respectively. Next, hp a posteriori error estimates are derived for the coupled state and control approximation. In the end, a projection gradient iterative algorithm is given, which solves the optimal control problems efficiently. Numerical experiments are carried out to confirm that the numerical results are in good agreement with the theoretical results.

     

Approximation of Singularities by Quantized-Tensor FEM

In d dimensions, first-order tensor-product finite-element (FE) approximations of the solutions of second-order elliptic problems are well known to converge algebraically, with rate at most 1/d in the energy norm and with respect to the number of degrees of freedom. On the other hand, FE methods of higher regularity may achieve exponential convergence, e.g. global spectral methods for analytic solutions and hp methods for solutions from certain countably normed spaces, which may exhibit singularities. In this note, we revisit, in one dimension, the tensor-structured approach to the h-FE approximation of singular functions. We outline a proof of the exponential convergence of such approximations represented in the quantized-tensor-train (QTT) format. Compared to special approximation techniques, such as hp, that approach is fully adaptive in the sense that it finds suitable approximation spaces algorithmically. The convergence is measured with respect to the number of parameters used to represent the solution, which is not the dimension of the first-order FE space, but depends only polylogarithmically on that. We demonstrate the convergence numerically for a simple model problem and find the rate to be approximately the same as for hp approximations. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

hp-Adaptive Finite Element Methods for Variational Inequalities

In this work, we combine an hp–adaptive strategy with a posteriori error estimates for variational inequalities, which are given by contact problems. The a posteriori error estimates are obtained using a general approach based on the saddle point formulation of contact problems and making use of a yposteriori error estimates for variational equations. Error estimates are presented for obstacle problems and Signorini problems with friction. Numerical experiments confirm the reliability of the error estimates for finite elements of higher order. The use of the hp–adaptive strategy leads to meshes with the same characteristics as geometric meshes and to exponential convergence. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On residual-based a posteriori error estimation in hp-FEM

A family , [0,1], of residual-based error indicators for the hp-version of the finite element method is presented and analyzed. Upper and lower bounds for the error indicators are established. To do so, the well-known Clément/Scott–Zhang interpolation operator is generalized to the hp-context and new polynomial inverse estimates are presented. An hp-adaptive strategy is proposed. Numerical examples illustrate the performance of the error indicators and the adaptive strategy.     

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.     

Convergence of an adaptive hp finite element strategy in higher space-dimensions

We show convergence in the energy norm for an automatic hp-adaptive refinement strategy for finite elements on the elliptic boundary value problem. The result is a generalization of a marking strategy, which was proposed in one space-dimension, to problems in two- and three-dimensional spaces.     

Spectral discretizations of 3-D elliptic problems and fast domain decomposition methods

An important for applications, the class of hp discretizations of second-order elliptic equations consists of discretizations based on spectral finite elements. The development of fast domain decomposition algorithms for them was restrained by the absence of fast solvers for the basic components of the method, i.e., for local interior problems on decomposition subdomains and their faces. Recently, the authors have established that such solvers can be designed using special factorized preconditioners. In turn, factorized preconditioners are constructed using an important analogy between the stiffness matrices of spectral and hierarchical basis hp-elements (coordinate functions of the latter are defined as tensor products of integrated Legendre polynomials). Due to this analogy, for matrices of spectral elements, fast solvers can be developed that are similar to those for matrices of hierarchical elements. Based on these facts and previous results on the preconditioning of other components, fast domain decomposition algorithms for spectral discretizations are obtained.     

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     

3D hp-adaptive finite element simulations of bend,step, and magic-T electromagnetic waveguide structures

Metallic rectangular waveguides are often the preferred choice on telecommunication systems and medical equipment working on the upper microwave and millimeter wave frequency bands due to the simplicity of its geometry, low losses, and the capacity to handle high powers. Waveguide translational symmetry is interrupted by the unavoidable presence of bends, transitions, and junctions, among others. This paper employs a 3D hp self-adaptive grid-refinement finite element strategy for the solution of these relevant electromagnetic waveguide problems. These structures often incorporate dielectrics, metallic screws, round corners, and so on, which may facilitate its construction or improve its design, but significantly difficult its modeling when employing semi-analytical techniques. The hp-adaptive finite element method enables accurate modeling of these structures even in the presence of complex materials and geometries. Numerical results demonstrate the suitability of the hp-adaptive method for modeling these waveguide structures, delivering errors below 0.5% with a limited number of unknowns. Solutions of waveguide problems obtained with the self-adaptive hp-FEM are of comparable accuracy to those obtained with semi-analytical techniques such as the Mode Matching method, for problems where the latest methods can be applied. At the same time, the hp-adaptive FEM enables accurate modeling of more complex waveguide structures.     

Exponential convergence in a Galerkin least squares hp-FEM for Stokes flow

A stabilized hp-finite element method (FEM) of Galerkin leastsquares (GLS) type is analysed for the Stokes equations in polygonaldomains. Contrary to the standard Galerkin FEM, the GLSFEM admitsthe implementationally attractive equal-order interpolationin the velocity and the pressure. In conjunction with geometricallyrefined meshes and linearly increasing approximation ordersit is shown that the hp-GLSFEM leads to exponential rates ofconvergence for solutions exhibiting singularities near corners.To obtain this result a novel hp-interpolation result is provedthat allows the approximation of pressure functions in certainweighted Sobolev spaces in a conforming way and at exponentialrates of convergence on geometric meshes. Received 6 June 1999. Accepted 14 March 2000.     

hp-Adaptivity in Two and Three Space-Dimensions

We present an hp-adaptive refinement strategy for two and three space-dimensions, which is based on the strategy presented in [2], and a convergence result for this strategy is stated. Then the refinement strategy is applied to some two- and three-dimensional model problems. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixed FEM of higher-order for a frictional contact problem

This paper presents mixed finite element methods of higher-order for an idealized frictional contact problem in linear elasticity. The approach relies on a saddle point formulation where the frictional contact condition is captured by a Lagrange multiplier. The convergence of the mixed scheme is proven and some a priori estimates for the h- and p-method are derived. Furthermore, a posteriori error estimates are presented which rely on the estimation of the discretization error of an auxiliary problem and some further terms capturing the error in the friction and complementary conditions. Numerical results confirm the applicability of the a posteriori error estimates within h- and hp-adaptive schemes. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical study of the HP version of mixed discontinuous finite element methods for reaction‐diffusion problems: The 1D case

This article studies the stability and convergence of the hp version of the three families of mixed discontinuous finite element (MDFE) methods for the numerical solution of reaction‐diffusion problems. The focus of this article is on these problems for one space dimension. Error estimates are obtained explicitly in the grid size h, the polynomial degree p, and the solution regularity; arbitrary space grids and polynomial degree are allowed. These estimates are asymptotically optimal in both h and p for some of these methods. Extensive numerical results to show convergence rates in h and p of the MDFE methods are presented. Theoretical and numerical comparisons between the three families of MDFE methods are described. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 525–553, 2003     

hp‐FEM for second moments of elliptic PDEs with stochastic data. II: Exponential convergence for stationary singular covariance functions

We prove exponential rates of convergence of a class of hp Galerkin Finite Element approximations of solutions to a model tensor nonhypoelliptic equation in the unit square □ = (0, 1)² which exhibit singularities on ?□ and on the diagonal Δ = {(x, y) ∈ □ : x = y}, but are otherwise analytic in □. As we explained in the first part (Pentenrieder and Schwab, Research Report, Seminar for Applied Mathematics, 2010) of this work, such problems arise as deterministic second moment equations of linear, second order elliptic operator equations Au = f with Gaussian random field data f. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

hp-Finite element for free vibration analysis of the orthotropic triangular and rectangular plates

The hp-version of the finite element method based on a triangular p-element is applied to free vibration of the orthotropic triangular and rectangular plates. The element's hierarchical shape functions, expressed in terms of shifted Legendre orthogonal polynomials, is developed for orthotropic plate analysis by taking into account shear deformation, rotary inertia, and other kinematics effects. Numerical results of frequency calculations are found for the free vibration of the orthotropic triangular and rectangular plates with the effect of the fiber orientation and plate boundary conditions. The results are very well compared to those presented in the literature.     

A posteriori finite element error estimators for indefinite elliptic boundary value problems ∗

A general construction technique is presented for a posteriori error estimators of finite element solutions of elliptic boundary value problems that satisfy a Gång inequality. The estimators are obtained by an element–by–element solution of ‘weak residual’ with or without considering element boundary residuals. There is no order restriction on the finite element spaces used for the approximate solution or the error estimation; that is, the design of the estimators is applicable in connection with either one of the h–p–, or hp– formulations of the finite element method. Under suitable assumptions it is shown that the estimators are bounded by constant multiples of the true error in a suitable norm. Some numerical results are given to demonstrate the effectiveness and efficiency of the approach.     

Two-Grid hp-Version DGFEMs for Strongly Monotone Second-Order Quasilinear Elliptic PDEs

In this article we develop the a priori error analysis of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical approximation of strongly monotone second-order quasilinear partial differential equations. In this setting, the fully nonlinear problem is first approximated on a coarse finite element space V(𝒯_H, P ). The resulting ‘coarse’ numerical solution is then exploited to provide the necessary data needed to linearize the underlying discretization on the finer space V(𝒯_h, p ); thereby, only a linear system of equations is solved on the richer space V(𝒯_h, p ). Numerical experiments confirming the theoretical results are presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interpolation error-based a posteriori error estimation for two-point boundary value problems and parabolic equations in one space dimension

Summary. I derive a posteriori error estimates for two-point boundary value problems and parabolic equations in one dimension based on interpolation error estimates. The interpolation error estimates are obtained from an extension of the error formula for the Lagrange interpolating polynomial in the case of symmetrically-spaced interpolation points. From this formula pointwise and seminorm a priori estimates of the interpolation error are derived. The interpolant in conjunction with the a priori estimates is used to obtain asymptotically exact a posteriori error estimates of the interpolation error. These a posteriori error estimates are extended to linear two-point boundary problems and parabolic equations. Computational results demonstrate the convergence of a posteriori error estimates and their effectiveness when combined with an hp-adaptive code for solving parabolic systems. Received April 17, 2000 / Revised version received September 25, 2000 / Published online May 30, 2001