Lowest Order Case Research Articles

Discrete Helmholtz Decompositions of Piecewise Constant and Piecewise Affine Vector and Tensor Fields

AbstractDiscrete Helmholtz decompositions dissect piecewise polynomial vector fields on simplicial meshes into piecewise gradients and rotations of finite element functions. This paper concisely reviews established results from the literature which all restrict to the lowest-order case of piecewise constants. Its main contribution consists of the generalization of these decompositions to 3D and of novel decompositions for piecewise affine vector fields in terms of Fortin–Soulie functions. While the classical lowest-order decompositions include one conforming and one nonconforming part, the decompositions of piecewise affine vector fields require a nonconforming enrichment in both parts. The presentation covers two and three spatial dimensions as well as generalizations to deviatoric tensor fields in the context of the Stokes equations and symmetric tensor fields for the linear elasticity and fourth-order problems. While the proofs focus on contractible domains, generalizations to multiply connected domains and domains with non-connected boundary are discussed as well.

A posteriori error estimates of the weak Galerkin finite element methods for parabolic problems

In this paper, We propose the residual-based a posteriori error estimator of the weak Galerkin finite element method with the backward Euler time discretization for the linear parabolic partial differential equation. For the a posteriori error estimator, we introduce the Helmholtz decomposition technique to prove its reliability. We mainly study WG element (Pj(K),Pℓ(∂K),V(K,r)=RTj(K)). Numerical experiments based on the lowest order case, i.e. (P0(K),P0(∂K),RT0(K)), are provided to verify the theoretical research.

Convergence of adaptive two-grid weak Galerkin finite element methods for semilinear elliptic differential equations

In this paper, we investigate the convergence of an adaptive two-grid weak Galerkin (ATGWG) finite element method for second order semilinear elliptic partial differential equations (PDEs). First, we propose an ATGWG method and then prove that the sum of the energy error and the error estimator of ATGWG method between two consecutive adaptive loops is a contraction. The weak Galerkin (WG) elements (Pj(T),Pℓ(∂T),RTj(T)) (Wang and Ye, 2013) are studied in this paper and numerical experiments based on the lowest order case with j=l=0 are provided to support the theoretical results.

An EMA-conserving, pressure-robust and Re-semi-robust method with A robust reconstruction method for Navier–Stokes

Proper EMA-balance (balance of kinetic energy, linear momentum and angular momentum), pressure-robustness and Re-semi-robustness (Re: Reynolds number) are three important properties of Navier–Stokes simulations with exactly divergence-free elements. This EMA-balance makes a method conserve kinetic energy, linear momentum and angular momentum in an appropriate sense; pressure-robustness means that the velocity errors are independent of the pressure; Re-semi-robustness means that the constants appearing in the error bounds of kinetic and dissipation energies do not explicitly depend on inverse powers of the viscosity. In this paper, based on the pressure-robust reconstruction framework and certain suggested reconstruction operators in Linke and Merdon [Comput. Methods Appl. Mech. Eng. 311 (2016) 304–326], we propose a reconstruction method for a class of non-divergence-free simplicial elements which admits almost all the above properties. The only exception is the energy balance, where kinetic energy should be replaced by a suitably redefined discrete energy. The lowest order case is the Bernardi–Raugel element on general shape-regular meshes. Some numerical comparisons with exactly divergence-free methods, the original pressure-robust reconstruction methods and the EMAC method are provided to confirm our theoretical results.

A lowest-order locking-free nonconforming virtual element method based on the reduced integration technique for linear elasticity problems

We develop a lowest-order nonconforming virtual element method for planar linear elasticity, which can be viewed as an extension of the idea in Falk (1991) to the virtual element method (VEM), with the family of polygonal meshes satisfying a very general geometric assumption. The method is shown to be uniformly convergent for the nearly incompressible case with optimal rates of convergence. The crucial step is to establish the discrete Korn's inequality, yielding the coercivity of the discrete bilinear form. We also provide a unified locking-free scheme both for the conforming and nonconforming VEMs in the lowest-order case. Numerical results validate the feasibility and effectiveness of the proposed numerical algorithms.

Higher-order adaptive virtual element methods with contraction properties

<abstract><p>The realization of a standard Adaptive Finite Element Method (AFEM) preserves the mesh conformity by performing a completion step in the refinement loop: In addition to elements marked for refinement due to their contribution to the global error estimator, other elements are refined. In the new perspective opened by the introduction of Virtual Element Methods (VEM), elements with hanging nodes can be viewed as polygons with aligned edges, carrying virtual functions together with standard polynomial functions. The potential advantage is that all activated degrees of freedom are motivated by error reduction, not just by geometric reasons. This point of view is at the basis of the paper [L. Beirão da Veiga et al., "Adaptive VEM: stabilization-free a posteriori error analysis and contraction property", SIAM Journal on Numerical Analysis, vol. 61, 2023], devoted to the convergence analysis of an adaptive VEM generated by the successive newest-vertex bisections of triangular elements without applying completion, in the lowest-order case (polynomial degree $ k = 1 $). The purpose of this paper is to extend these results to the case of VEMs of order $ k\ge2 $ built on triangular meshes. The problem at hand is a variable-coefficient, second-order self-adjoint elliptic equation with Dirichlet boundary conditions; the data of the problem are assumed to be piecewise polynomials of degree $ k-1 $. By extending the concept of global index of a hanging node, under an admissibility assumption of the mesh, we derive a stabilization-free a posteriori error estimator. This is the sum of residual-type terms and certain virtual inconsistency terms (which vanish for $ k = 1 $). We define an adaptive VEM of order $ k $ based on this estimator, and we prove its convergence by establishing a contraction result for a linear combination of (squared) energy norm of the error, (squared) residual estimator, and (squared) virtual inconsistency estimator.</p></abstract>

A discrete de Rham method for the Reissner–Mindlin plate bending problem on polygonal meshes

In this work we propose a discretisation method for the Reissner–Mindlin plate bending problem in primitive variables that supports general polygonal meshes and arbitrary order. The method is inspired by a two-dimensional discrete de Rham complex for which key commutation properties hold that enable the cancellation of the contribution to the error linked to the enforcement of the Kirchhoff constraint. Denoting by k≥0 the polynomial degree for the discrete spaces and by h the meshsize, we derive for the proposed method an error estimate in hk+1 for general k, as well as a locking-free error estimate for the lowest-order case k=0. The theoretical results are validated on a complete panel of numerical tests.

Exact sequences on Worsey–Farin splits

We construct several smooth finite element spaces defined on three-dimensional Worsey–Farin splits. In particular, we construct C 1 C^1 , H 1 ( curl ) H^1(\operatorname {curl}) , and H 1 H^1 -conforming finite element spaces and show the discrete spaces satisfy local exactness properties. A feature of the spaces is their low polynomial degree and lack of extrinsic supersmoothness at subsimplices of the mesh. In the lowest order case, the last two spaces in the sequence consist of piecewise linear and piecewise constant spaces, and are suitable for the discretization of the (Navier-)Stokes equation.

High-order space–time finite element methods for the Poisson–Nernst–Planck equations: Positivity and unconditional energy stability

We present a novel class of high-order space–time finite element schemes for the Poisson–Nernst–Planck (PNP) equations. We prove that our schemes are mass conservative, positivity preserving, and unconditionally energy stable for any order of approximation. To the best of our knowledge, this is the first class of (arbitrarily) high-order accurate schemes for the PNP equations that simultaneously achieve all these three properties.This is accomplished via (1) using finite elements to directly approximate the so-called entropy variableui≔U′(ci)=log(ci) instead of the density variable ci, where U(ci)=(log(ci)−1)ci is the corresponding entropy, and (2) using a discontinuous Galerkin (DG) discretization in time. The entropy variable formulation, which was originally developed by Metti et al. (2016) under the name of a log-density formulation, guarantees both positivity of densities ci=exp(ui)>0 and a continuous-in-time energy stability result. The DG in time discretization further ensures an unconditional energy stability in the fully discrete level for any approximation order, where the lowest order case is exactly the backward Euler discretization and in this case we recover the method of Metti et al. (2016).

A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

<p style='text-indent:20px;'>The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problem and a set of representative polygonal meshes. We numerically show that this formulation is convergent with optimal convergence rates except for the lowest-order case on triangular meshes, where the method coincides with the <inline-formula><tex-math id="M1">\begin{document}$ {\mathbb{P}}_{{1}}-{\mathbb{P}}_{{0}} $\end{document}</tex-math></inline-formula> Scott-Vogelius scheme, and on square meshes, which are situations that are well-known to be unstable.</p>

The curl–curl conforming virtual element method for the quad-curl problem

In this paper, we present the [Formula: see text]-conforming virtual element (VE) method for the quad-curl problem in two dimensions. Based on the idea of de Rham complex, we first construct three families of [Formula: see text]-conforming VEs, of which the simplest one has only one degree of freedom associated to each vertex and each edge in the lowest-order case, respectively. An exact discrete complex is established between the [Formula: see text]-conforming and [Formula: see text]-conforming VEs. We rigorously prove the interpolation error estimates, the stability of discrete bilinear forms, the coercivity and inf–sup condition of the corresponding discrete formulation. We show that the conforming VEs have the optimal convergence. Some numerical examples are given to confirm the theoretical results.

A New Projection-Based Stabilized Virtual Element Method for the Stokes Problem

We propose and analyze a stabilized virtual element method for the Stokes problem on polytopal meshes. We employ the $$C^0$$ continuous arbitrary “equal-order” virtual element pairs to approximate both velocity and pressure, and develop a projection-based stabilization term to circumvent the discrete inf-sup condition, then we obtain the corresponding error estimates. The presented method involves neither the projection of the second derivative nor additional coupling terms, and it is parameter-free. In particularly, for the lowest-order case on triangular (tetrahedral) meshes the stabilized method introduced by Bochev et al. (SIAM J. Numer. Anal. 44: 82–101, 2006) is a special case of our method up to an approximation of the load term. Furthermore, numerical results are shown to confirm the theoretical predictions.

The nonconforming virtual element method for the Darcy–Stokes problem

We present the nonconforming virtual element method for the Darcy–Stokes problem. By using a gradient projection operator and a special polynomial subspace orthogonal to the gradient of some polynomial space, we construct a H1-nonconforming but H(div)-conforming virtual element that allows us to compute the L2-projection. The optimal convergence is proved under the assumption of sufficient regularity, and the uniform convergence is also obtained for the lowest-order case. Besides, we establish a discrete exact sequence of de Rham complex related to the nonconforming virtual element. Finally, we carry out some numerical tests to make a comparison of the convergence between different nonconforming virtual elements and confirm the optimal and uniform convergence of the nonconforming virtual element.

Conforming and nonconforming virtual element methods for a Kirchhoff plate contact problem

AbstractWe establish a general framework to study the conforming and nonconforming virtual element methods (VEMs) for solving a Kirchhoff plate contact problem with friction, which is a fourth-order elliptic variational inequality (VI) of the second kind. This VI contains a non-differentiable term due to the frictional contact. This theoretical framework applies to the existing virtual elements such as the conforming element, the $C^0$-continuous nonconforming element and the fully nonconforming Morley-type element. In the unified framework we derive a priori error estimates for these virtual elements and show that they achieve optimal convergence order for the lowest-order case. For demonstrating the performance of the VEMs we present some numerical results that confirm the theoretical prediction of the convergence order.

Moments of the Dedekind zeta function and other non-primitiveL-functions

AbstractWe give a conjecture for the moments of the Dedekind zeta function of a Galois extension. This is achieved through the hybrid product method of Gonek, Hughes and Keating. The moments of the product over primes are evaluated using a theorem of Montgomery and Vaughan, whilst the moments of the product over zeros are conjectured using a heuristic method involving random matrix theory. The asymptotic formula of the latter is then proved for quadratic extensions in the lowest order case. We are also able to reproduce our moments conjecture in the case of quadratic extensions by using a modified version of the moments recipe of Conrey et al. Generalising our methods, we then provide a conjecture for moments of non-primitiveL-functions, which is supported by some calculations based on Selberg’s conjectures.

First-order least-squares method for the obstacle problem

We define and analyse a least-squares finite element method for a first-order reformulation of the obstacle problem. Moreover, we derive variational inequalities that are based on similar but non-symmetric bilinear forms. A priori error estimates including the case of non-conforming convex sets are given and optimal convergence rates are shown for the lowest-order case. We provide a posteriori bounds that can be used as error indicators in an adaptive algorithm. Numerical studies are presented.

A [formula omitted] Virtual Element Method on polyhedral meshes

The purpose of the present paper is to develop C1 Virtual Elements in three dimensions for linear elliptic fourth order problems, motivated by the difficulties that standard conforming Finite Elements encounter in this framework. We focus the presentation on the lowest order case, the generalization to higher orders being briefly provided in the Appendix. The degrees of freedom of the proposed scheme are only 4 per mesh vertex, representing function values and gradient values. Interpolation error estimates for the proposed space are provided, together with a set of numerical tests to validate the method at the practical level.

Full [formula omitted]-approximation of linear elasticity on quadrilateral meshes based on [formula omitted] finite elements

For meshes of nondegenerate, convex quadrilaterals, we present a family of stable mixed finite element spaces for the mixed formulation of planar linear elasticity. The problem is posed in terms of the stress tensor, the displacement vector and the rotation scalar fields, with the symmetry of the stress tensor weakly imposed. The proposed spaces are based on the Arnold–Boffi–Falk (ABFk, k≥0) elements for the stress and piecewise polynomials for the displacement and the rotation. We prove that these finite elements provide full H(div)-approximation of the stress field, in the sense that it is approximated to order hk+1, where h is the mesh diameter, in the H(div)-norm. We show that displacement and rotation are also approximated to order hk+1 in the L2-norm. The convergence is optimal order for k≥1, while the lowest order case, index k=0, requires special treatment. The spaces also apply to both compressible and incompressible isotropic problems, i.e., the Poisson ratio may be one-half. The implementation as a hybrid method is discussed, and numerical results are given to illustrate the effectiveness of these finite elements.

The nonconforming virtual element method for elasticity problems

We present the nonconforming virtual element method for linear elasticity problems in the pure displacement formulation. This method is uniformly convergent for the nearly incompressible case. The optimal convergence in the H1 norm is proved under some regularity assumptions. We mention that, for the lowest-order case on triangular meshes the nonconforming virtual element method coincides with the nonconforming finite element method presented by Brenner and Sung (1992) [19] up to an approximation of the right-hand side. Besides, we also present the nonconforming virtual element method for the pure traction problem, which is proved to be uniformly convergent with respect to the Lamé constant. Finally, we demonstrate the optimal convergence and stability of the nonconforming virtual element method by some numerical results.

The Morley-Type Virtual Element for Plate Bending Problems

We propose a simple nonconforming virtual element for plate bending problems, which has few local degrees of freedom and provides the optimal convergence in $$H^2$$ -norm. Moreover, we prove the optimal error estimates in $$H^1$$ - and $$L^2$$ -norm. The nonconforming virtual element is constructed for any order of accuracy, but not $$C^0$$ -continuous. It is worth mentioning that, for the lowest-order case on triangular meshes the simplified nonconforming virtual element coincides with the well-known Morley element, so it can be taken as the extension of the Morley element to polygonal meshes. Finally, we verify the optimal convergence in $$H^2$$ -norm for the nonconforming virtual element by some numerical tests.