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We construct three H-curl-curl finite elements. The P2P_{2}and P3P_{3}vector finite element spaces are both enriched by one common P4P_{4}bubble and their local degrees of freedom are 13 and 21, respectively. As there does not exist any P1P_{1}H-curl-curl conforming finite element, the P1P_{1}H-curl-curl nonconforming finite element is constructed with three additional P4P_{4}bubbles. Numerical tests are presented, confirming the conformity and the optimal order of convergence.

We construct a C1C^{1}-P7P_{7}Bell finite element by restricting its normal derivative from a P6P_{6}polynomial to a P5P_{5}polynomial, and its second normal derivative from a P5P_{5}polynomial to a P4P_{4}polynomial, on the three edges of every triangle. On one triangle, the finite element space contains the P6P_{6}polynomial space. We show the method converges at order 7 in L2L^{2}-norm. By eliminating all degrees of freedom on edges of C1C^{1}-P7P_{7}Argyris finite element, the global degrees of freedom of the new element are reduced substantially from 27⁢V27Vto 12⁢V12Vasymptotically, where 𝑉 is the number of vertices in the triangular mesh. While the global degrees of freedom of the C1C^{1}-P6P_{6}Argyris finite element is 19⁢V19V, the new element is equally accurate but more economic. Numerical tests are presented, showing the new element is more accurate than the existing element while having less global unknowns.

A family of conforming finite elements are designed on rectangular grids for solving the Reissner-Mindlin plate equation. The rotation is approximated by C1-Qk+1in one direction and C0-Qkin the other direction finite elements. The displacement is approximated by C1-Qk+1,k+1. The method is locking-free without using any projection/reduction operator. Theoretical proof and numerical confirmation are presented.

The continuous P3and discontinuous P2finite element pair is stable on subquadrilateral triangular meshes for solving 2D stationary Stokes equations. By putting two diagonal lines into every quadrilateral of a quadrilateral mesh, we get a subquadrilateral triangular mesh. Such a velocity solution is divergence-free point wise and viscosity robust in the sense the solution and the error are independent of the viscosity. Numerical examples show an advantage of such a method over the Taylor-Hood P3-P2method, where the latter deteriorates when the viscosity becomes small.

A novel conforming discontinuous Galerkin (CDG) finite element method is introduced for Poisson equation on rectangular meshes. This CDG method with discontinuous Pk(k⩾1) elements converges to the true solution two orders above the continuous finite element counterpart. Superconvergence of order two for the CDG finite element solution is proved in an energy norm and the L2norm. A local post-process is defined which lifts a PkCDG solution to a discontinuous Pk+2solution. It is proved that the lifted Pk+2solution converges at the optimal order. The numerical tests illustrate the theoretic findings.

The family of Falk-Neilan Pkfinite elements, combined with the Argyris Pk+1finite elements, solves the Reissner-Mindlin plate equation quasi-optimally and locking-free, on triangular meshes. The method is truly conforming or consistent in the sense that no projection/reduction is introduced. Theoretical proof and numerical confirmation are presented.

A stabilizer-free weak Galerkin (SFWG) finite element method was introduced and analyzed in Ye and Zhang (SIAM J. Numer. Anal. 58: 2572–2588, 2020) for the biharmonic equation, which has an ultra simple finite element formulation. This work is a continuation of our investigation of the SFWG method for the biharmonic equation. The new SFWG method is highly accurate with a convergence rate of four orders higher than the optimal order of convergence in both the energy norm and the L2norm on triangular grids. This new method also keeps the formulation that is symmetric, positive definite, and stabilizer-free. Four-order superconvergence error estimates are proved for the corresponding SFWG finite element solutions in a discrete H2norm. Superconvergence of four orders in the L2norm is also derived for k⩾3, where kis the degree of the approximation polynomial. The postprocessing is proved to lift a PkSFWG solution to a Pk+4solution elementwise which converges at the optimal order. Numerical examples are tested to verify the theories.

We consider the mixed discontinuous Galerkin (DG) finite element approximation of the Stokes equation and provide the analysis for the [Pk]d-Pk-1element on the tensor product meshes. Comparing to the previous stability proof with [Qk]d-Qk-1discontinuous finite elements in the existing references, our first contribution is to extend the formal proof to the [Pk]d-Pk-1discontinuous elements on the tensor product meshes. Numerical inf-sup tests have been performed to compare Qkand Pktypes of elements and validate the well-posedness in both settings. Secondly, our contribution is to design the enhanced pressure-robust discretization by only modifying the body source assembling on [Pk]d-Pk-1schemes to improve the numerical simulation further. The produced numerical velocity solution via our enhancement shows viscosity and pressure independence and thus outperforms the solution produced by standard discontinuous Galerkin schemes. Robustness analysis and numerical tests have been provided to validate the scheme’s robustness.

This article extends a recently developed superconvergence result for weak Galerkin (WG) approximations for modeling partial differential equations from constant coefficients to variable coefficients. This superconvergence features a rate that is two-order higher than the optimal-order error estimates in the usual energy and L2norms. The extension from constant to variable coefficients for the modeling equations is highly non-trivial. The underlying technical analysis is based on the use of a sequence of projections and decompositions. Numerical results are presented to confirm the superconvergence theory for second-order elliptic problems with variable coefficients.

In this paper, we introduce an H(div)finite element method on polygonal and polyhedral meshes for solving the Stokes equations in the primary velocity–pressure formulation. An H(div)finite element on polygons or polyhedra is introduced to approximate the velocity so that the method is pressure robust and produces exact divergence free solutions, when the discontinuous Pkfinite element is adopted for the pressure approximation. In addition, this method is stabilizer-free, in handling the H1non-conformity. Optimal order error estimates are established for the method. Numerical tests are conducted to confirm the theory.

In this paper, we introduce new stable mixed finite elements of any order on polytopal mesh for solving second-order elliptic problem. We establish optimal order error estimates for velocity and super convergence for pressure. Numerical experiments are conducted for our mixed elements of different orders on 2D and 3D spaces that confirm the theory.

In this paper, we propose a new numerical scheme for the coupled Stokes-Darcy model with the Beavers-Joseph-Saffman interface condition. We use the weak Galerkin method to discretize the Stokes equation and the mixed finite element method to discretize the Darcy equation. A discrete inf-sup condition is proved and the optimal error estimates are also derived. Numerical experiments validate the theoretical analysis.

The lowest degree of polynomial for a finite element to solve a 2kth-order elliptic equation is k. The Morley element is such a finite element, of polynomial degree 2, for solving a fourth-order biharmonic equation. We design a cubic $$H^3$$H3-nonconforming macro-element on two-dimensional triangular grids, solving a sixth-order tri-harmonic equation. We also write down explicitly the 12 basis functions on each macro-element. A convergence theory is established and verified by numerical tests.

AbstractIn this paper, we propose to use the interior functions of an hierarchical basis for high order BDMpelements to enforce the divergence-free condition of a magnetic field Bapproximated by the H(div)BDMpbasis. The resulting constrained finite element method can be used to solve magnetic induction equation in MHD equations. The proposed procedure is based on the fact that the scalar (p–1)-th order polynomial space on each element can be decomposed as an orthogonal sum of the subspace defined by the divergence of the interior functions of the p-th order BDMpbasis and the constant function. Therefore, the interior functions can be used to remove element-wise all higher order terms except the constant in the divergence error of the finite element solution of the B-field. The constant terms from each element can be then easily corrected using a first order H(div)basis globally. Numerical results for a 3-D magnetic induction equation show the effectiveness of the proposed method in enforcing divergence-free condition of the magnetic field.

A new interpolation operator is defined, which preserves only P2 polynomials locally.

A family of Pk+22×2-PkH(divdiv)mixed finite elements on triangular grids is constructed for the biharmonic equation. For the H(divdiv)mixed finite element functions, both the stress tensor and its divergence have continuous normal components, σh⋅nand divσh⋅n, on each edge. The double divergence of the H(divdiv)mixed finite element space is exact the full discontinuous Pkpolynomial space on a triangular grid. The stability of the mixed method is established. The stress solution converges at the optimal order in L2and H(divdiv)norms. Four-order of superconvergence is proved in L2norm for the displacement solution. The Pkdiscrete solution is processed on each triangle to obtain an optimal order Pk+4solution there. Numerical results are presented verifying the theory. Additional computation shows that all existing H(divdiv)mixed finite elements fail to solve jump-coefficient biharmonic equations while the new mixed element converges independently of the interface-jump. This is because the H(divdiv)element here is the full H(divdiv)−Pk+2space while the existing mixed elements are subspaces of the H(divdiv)−Pk+2space, on a triangular grid.

A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed, where the stress is approximated by symmetric H(div)-Pkpolynomial tensors and the displacement is approximated by C−1-Pk−1polynomial vectors, for all k⩽ 4. The main ingredients for the analysis are a new basis of the space of symmetric matrices, an intrinsic H(div) bubble function space on each element, and a new technique for establishing the discrete inf-sup condition. In particular, they enable us to prove that the divergence space of the H(div) bubble function space is identical to the orthogonal complement space of the rigid motion space with respect to the vector-valued Pk−1polynomial space on each tetrahedron. The optimal error estimate is proved, verified by numerical examples.

It is well known that it is comparatively difficult to design nonconforming finite elements on quadrilateral meshes by using Gauss-Legendre points on each edge of triangulations. One reason lies in that these degrees of freedom associated with these Gauss-Legendre points are not all linearly independent for usual expected polynomial spaces, which explains why only several lower order nonconforming quadrilateral finite elements can be found in literature. The present paper proposes two families of nonconforming finite elements of any odd order and one family of nonconforming finite elements of any even order on quadrilateral meshes. Degrees of freedom are given for these elements, which are proved to be well-defined for their corresponding shape function spaces in a unifying way. These elements generalize three lower order nonconforming finite elements on quadrilaterals to any order. In addition, these nonconforming finite element spaces are shown to be full spaces which is somehow not discussed for nonconforming finite elements in literature before.

By the standard theory, the stable Qk+1,k−Qk,k+1/$Q_{k}^{dc'}$divergence-free element converges with the optimal order of approximation for the Stokes equations, but only order kfor the velocity in H1-norm and the pressure in L2-norm. This is due to one polynomial degree less in ydirection for the first component of velocity, which is a Qk+1,kpolynomial of xand y. In this manuscript, we will show by supercloseness of the divergence free element that the order of convergence is truly k+1, for both velocity and pressure. For special solutions (if the interpolation is also divergence-free), a two-order supercloseness is shown to exist. Numerical tests are provided confirming the accuracy of the theory.

The distributed relaxation method for the Stokes problem has been advertised as an adequate change of variables that leads to a lower triangular system with Laplace operators on the main diagonal for which multigrid methods are very efficient. We show that under high regularity of the Laplacian, the transformed system admits almost block‐lower triangular form. We analyze the distributed relaxation method and compare it with other iterative methods for solving the Stokes system. We also present numerical experiments illustrating the effectiveness of the transformation which is well established for certain finite difference discretizations of Stokes problems. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 898–914, 2011

Summary. It is shown in this paper that the optimal approximation property of bilinear or trilinear finite elements would be retained when the affine mapping is used to replace the Q 1 mapping on each element, if the grids are refined nestedly. The new method truncates the quadratic and cubic terms in reference mappings and produces constant Jacobians and Jacobian matrices. This would avoid a shortcoming of the quadrilateral and hexahedral elements where the integrals of rational functions have to be computed or approximated. Numerical tests verify the analysis.

In this paper, we consider several model problems where finite element triangular meshes with arbitrarily small angles (high aspect ratios) are utilized to deal with anisotropy, interfaces, or singular perturbations. The constant-rate (independent of the number of unknowns, the smallest angle, the interface discontinuity, the singular-perturbation parameter, etc.) convergence of some special nonnested multigrid methods for solving the finite element systems on such degenerate meshes will be proved. Numerical data are provided to support the analysis in each case.

We prove that the multigrid method works with optimal computational order even when the multiple meshes are not nested. When a coarse mesh is not a submesh of the finer one, the coarse-level correction usually does not have the $ a( \cdot , \cdot )$V-cycle or a W-cycle method. It is shown numerically that the convergence rates of nonnested multigrid methods are not necessarily worse than those of nested ones. Since nonnested multigrid methods accept quite arbitrarily related meshes, we may then combine the efficiencies of adaptive refinements and of multigrid algorithms.

In this paper, we propose a modified Lagrange type interpolation operator to approximate functions in Sobolev spaces by continuous piecewise polynomials. In order to define interpolators for "rough" functions and to preserve piecewise polynomial boundary conditions, the approximated functions are averaged appropriately either on d- or -simplices to generate nodal values for the interpolation operator. This combination of averaging and interpolation is shown to be a projection, and optimal error estimates are proved for the projection error.

Nonnested multigrid methods are proved to be optimal-order solvers for finite element equations arising from elliptic problems in the presence of singularities caused by re-entrant corners and abrupt changes in the boundary conditions, where the multilevel grids are appropriately refined near singularities and are not necessarily nested. Therefore, optimal and realistic finer grids (compared with nested local refinements) could be used because of the freedom in generating nonnested multilevel grids.

Nonnested multigrid methods are shown to be optimal-order solvers for systems of finite element equations arising from elliptic boundary problems in any space dimension. Results are derived for Lagrange-type elements of arbitrary degree.

Summary In this paper, we study a special multigrid method for solving large linear systems which arise from discretizing biharmonic problems by the Hsieh-Clough-Tocher,C1 macro finite elements or several otherC1 finite elements. Since the multipleC1 finite element spaces considered are not nested, the nodal interpolation operator is used to transfer functions between consecutive levels in the multigrid method. This method converges with the optimal computational order.

In this paper, we study the stabilizer-free weak Galerkin methods on polytopal meshes for a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient. With certain assumptions on the nonlinear coefficient, we show that the discrete problem has a unique solution. This is achieved by showing that the associated operator satisfies certain continuity and monotonicity properties. With the help of these properties, we derive optimal error estimates in the energy norm. We present several numerical examples to verify the error estimates.