Higher-order Finite Element Spaces Research Articles

High order approximations of the operator Lyapunov equation have low rank

We present a low-rank greedily adapted hp-finite element algorithm for computing an approximation to the solution of the Lyapunov operator equation. We show that there is a hidden regularity in eigenfunctions of the solution of the Lyapunov equation which can be utilized to justify the use of high order finite element spaces. Our numerical experiments indicate that we achieve eight figures of accuracy for computing the trace of the solution of the Lyapunov equation posed in a dumbbell-domain using a finite element space of dimension of only 10^4 degrees of freedom. Even more surprising is the observation that hp-refinement has an effect of reducing the rank of the approximation of the solution.

Hierarchical high order finite element spaces in [formula omitted] for a stabilized mixed formulation of Darcy problem

The classical dual mixed finite element method for flow simulations is based on H(div,Ω) conforming approximation spaces for the flux, which guarantees continuous normal components on element interfaces, and discontinuous approximations in L2(Ω) for the pressure. However, stability and convergence can only be obtained for compatible approximation spaces. Stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance. The main purpose of this paper is to present a high-order finite element methodology to solve the Darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of H(div,Ω) and H1(Ω). The chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables, without requiring any compatibility constraint, as required in mixed methods for these problems. Convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with Raviart–Thomas elements.

An Oseen-Type Post-Processed Finite Element Method Based on a Subgrid Model for the Time-Dependent Navier–Stokes Equations

An Oseen-type post-processed mixed finite element method based on a subgrid model is presented for the simulation of time-dependent incompressible Navier–Stokes equations. This method first solves a subgrid stabilized nonlinear Navier–Stokes system on a mesh of size [Formula: see text] to obtain an approximate solution pair [Formula: see text] at the given final time [Formula: see text], and then post-processes the solution [Formula: see text] by solving a stabilized Oseen problem on a finer mesh or in higher-order finite element spaces. We prove stability of the stabilized method, derive error estimates for the post-processed solutions, give some numerical results to verify the theoretical predictions and demonstrate the effectiveness of the proposed method.

Discrete Total Variation with Finite Elements and Applications to Imaging

The total variation (TV)-seminorm is considered for piecewise polynomial, globally discontinuous (DG) and continuous (CG) finite element functions on simplicial meshes. A novel, discrete variant (DTV) based on a nodal quadrature formula is defined. DTV has favorable properties, compared to the original TV-seminorm for finite element functions. These include a convenient dual representation in terms of the supremum over the space of Raviart–Thomas finite element functions, subject to a set of simple constraints. It can therefore be shown that a variety of algorithms for classical image reconstruction problems, including TV- $$L^2$$ denoising and inpainting, can be implemented in low- and higher-order finite element spaces with the same efficiency as their counterparts originally developed for images on Cartesian grids.

Convergence and Optimality of Higher-Order Adaptive Finite Element Methods for Eigenvalue Clusters

Proofs of convergence of adaptive finite element methods (AFEMs) for approximating eigenvalues and eigenfunctions of linear elliptic problems have been given in several recent papers. A key step in establishing such results for multiple and clustered eigenvalues was provided by Dai, He, and Zhou in [IMA J. Numer. Anal., 35 (2015), pp. 1934--1977], who proved convergence and optimality of AFEMs for eigenvalues of multiplicity greater than one. There it was shown that a theoretical (noncomputable) error estimator for which standard convergence proofs apply is equivalent to a standard computable estimator on sufficiently fine grids. In [Numer. Math., 130 (2015), pp. 467--496], Gallistl used a similar tool to prove that a standard AFEM for controlling eigenvalue clusters for the Laplacian using continuous piecewise linear finite element spaces converges with optimal rate. When considering either higher-order finite element spaces or nonconstant diffusion coefficients, however, the arguments of Dai, He, and Zhou and Gallistl do not yield equivalence of the practical and theoretical estimators for clustered eigenvalues. In this article we provide this missing key step, thus showing that standard adaptive FEMs for clustered eigenvalues employing elements of arbitrary polynomial degree converge with optimal rate. We additionally establish that a key user-defined input parameter in the AFEM, the bulk marking parameter, may be chosen entirely independently of the properties of the target eigenvalue cluster. All of these results assume a fineness condition on the initial mesh in order to ensure that the nonlinearity is sufficiently resolved.

Entropy–viscosity method for the single material Euler equations in Lagrangian frame

A new finite element method for solving the Euler equations in Lagrangian coordinates is proposed. The method is stabilized by adding artificial diffusion terms compatible with positivity of mass and internal energy, a minimum principle on the specific entropy, and all generalized entropy inequalities. Two options of first-order artificial diffusion are considered. One is in the spirit of the Eulerian based method (Guermond et al., 2011 [23, 22]; Zingan et al., 2013) and the other is similar to existing viscosity stabilizations in Lagrangian frame, e.g., Dobrev et al. (2012). The method is verified to be high-order for smooth solutions even with active viscosity terms. This is achieved by using high-order finite element spaces and an entropy-based viscosity stabilization that degenerates the first-order viscous terms. This stabilization automatically distinguishes smooth and singular regions. The formal accuracy and convergence properties of the proposed methods are tested on a series of benchmark problems. This is the first result extending the entropy–viscosity methodology to the Lagrangian hydrodynamics.

On high order finite element spaces of differential forms

We show how the high order finite element spaces of differential forms due to Raviart-Thomas-Nedelec-Hiptmair fit into the framework of finite element systems, in an elaboration of the finite element exterior calculus of Arnold-Falk-Winther. Based on observations by Bossavit, we provide new low order degrees of freedom. As an alternative to existing choices of bases, we provide canonical resolutions in terms of scalar polynomials and Whitney forms.

Towards a complete FEM-based simulation toolkit on GPUs: Unstructured grid finite element geometric multigrid solvers with strong smoothers based on sparse approximate inverses

We describe our FE-gMG solver, a finite element geometric multigrid approach for problems relying on unstructured grids. We augment our GPU- and multicore-oriented implementation technique based on cascades of sparse matrix–vector multiplication by applying strong smoothers. In particular, we employ Sparse Approximate Inverse (SPAI) and Stabilised Approximate Inverse (SAINV) techniques. We focus on presenting the numerical efficiency of our smoothers in combination with low- and high-order finite element spaces as well as the hardware efficiency of the FE-gMG. For a representative problem and computational grids in 2D and 3D, we achieve a speedup of an average of 5 on a single GPU over a multithreaded CPU code in our benchmarks. In addition, our strong smoothers can deliver a speedup of 3.5 depending on the element space, compared to simple Jacobi smoothing. This can even be enhanced to a factor of 7 when combining the usage of approximate inverse-based smoothers with clever sorting of the degrees of freedom. In total the FE-gMG solver can outperform a simple (multicore-) CPU-based multigrid by a total factor of over 40.

Higher Order Mortar Finite Element Methods in 3D with Dual Lagrange Multiplier Bases

Mortar methods with dual Lagrange multiplier bases provide a flexible, efficient and optimal way to couple different discretization schemes or nonmatching triangulations. Here, we generalize the concept of dual Lagrange multiplier bases by relaxing the condition that the trace space of the approximation space at the slave side with zero boundary condition on the interface and the Lagrange multiplier space have the same dimension. We provide a new theoretical framework within this relaxed setting, which opens a new and simpler way to construct dual Lagrange multiplier bases for higher order finite element spaces. As examples, we consider quadratic and cubic tetrahedral elements and quadratic serendipity hexahedral elements. Numerical results illustrate the performance of our approach.

Higher Order Triangular Finite Elements with Mass Lumping for the Wave Equation

In this article, we construct new higher order finite element spaces for the approximation of the two-dimensional (2D) wave equation. These elements lead to explicit methods after time discretization through the use of appropriate quadrature formulas which permit mass lumping. These formulas are constructed explicitly. Error estimates are provided for the corresponding semidiscrete problem. Finally, higher order finite difference time discretizations are proposed and various numerical results are shown.

Hierarchical A Posteriori Error Estimators for Mortar Finite Element Methods with Lagrange Multipliers

Hierarchical a posteriori error estimators are introduced and analyzed for mortar finite element methods. A weak continuity condition at the interfaces is enforced by means of Lagrange multipliers. The two proposed error estimators are based on a defect correction in higher-order finite element spaces and an adequate hierarchical two-level splitting. The first provides upper and lower bounds for the discrete energy norm of the mortar finite element solution whereas the second also estimates the error for the Lagrange multiplier. It is shown that an appropriate measure for the nonconformity of the mortar finite element solution is the weighted L2 -norm of the jumps across the interfaces.

Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates

This part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in R N \mathbb {R}^{N} . In a sense to be discussed below these sharpen known quasi–optimal L ∞ L_{\infty } and W ∞ 1 W^{1}_{\infty } estimates for the error on irregular quasi–uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u u . We note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces. As a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities. These expansion inequalities are valid for large classes of finite elements defined on irregular grids in R N \mathbb {R}^{N} and have applications to superconvergence and extrapolation and a posteriori estimates. Part II of this series will contain local estimates applicable to non–smooth problems.