General Quadrilateral Meshes Research Articles

Approximation properties of primal discontinuous Petrov-Galerkin method on general quadrilateral meshes

We consider the approximation properties of primal discontinuous Petrov-Galerkin (DPG) method on quadrilateral meshes. We show how the previous convergence results as well as the Aubin-Nitsche type duality arguments can be extended to cover arbitrary convex quadrilateral elements with bilinear isomorphisms. The arguments are based on the approximation theory of quadrilateral vector finite element spaces associated to the numerical flux variable of the DPG approximation. The theoretical results are validated by a numerical experiment that features also a comparison between the primal DPG method and a conventional least squares finite element method with the same number of degrees of freedom.

Neural-Network-Assisted Finite Difference Discretization for Numerical Solution of Partial Differential Equations

A neural-network-assisted numerical method is proposed for the solution of Laplace and Poisson problems. Finite differences are applied to approximate the spatial Laplacian operator on nonuniform grids. For this, a neural network is trained to compute the corresponding coefficients for general quadrilateral meshes. Depending on the position of a given grid point x0 and its neighbors, we face with a nonlinear optimization problem to obtain the finite difference coefficients in x0. This computing step is executed with an artificial neural network. In this way, for any geometric setup of the neighboring grid points, we immediately obtain the corresponding coefficients. The construction of an appropriate training data set is also discussed, which is based on the solution of overdetermined linear systems. The method was experimentally validated on a number of numerical tests. As expected, it delivers a fast and reliable algorithm for solving Poisson problems.

A finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes

We construct a finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes, where the strict convexity restriction on the meshes is removed. The main contributions of this paper include three aspects. Firstly, the classical finite volume element (C-FVE) method is extended to severely distorted quadrilateral meshes even with concave cells by virtue of a new overlapping dual partition and a new gradient approximation. In fact, the choice of this dual partition and gradient approximation is also conducive to the construction of a monotone scheme. Secondly, a new monotonicity correction is suggested, based on which we obtain a monotone finite volume element (M-FVE) method. The resulting M-FVE method still keeps the local conservation and is easy for implementation. Finally, we analyze theoretically the truncation error and the monotonicity for this scheme. Besides, the existence of a solution to this nonlinear scheme is proved by applying the Brouwer's fixed point theorem. Numerical results demonstrate that the M-FVE method has the approximate second-order accuracy and preserves well the positivity of the solution for both isotropic and anisotropic diffusion problems on severely distorted quadrilateral meshes.

Lid driven flow field statistics: A non-conforming finite element Simulation

In this paper, we implement the non-conforming finite elements for the simulation of two dimensional incompressible Newtonian fluid flows. The classical driven cavity benchmark problem is simulated on a very fine uniform grid. In order to discretize the governing partial differential equations (PDEs) we employ the rotated bilinearQ˜1element for velocity and piecewise constant Q0element for the pressure on the general quadrilateral meshes. In L2-norm this pair is 2ndorder accurate for velocity and provides 1st order accuracy for pressure approximations. For the solution of non-linear systems arising from the discretization, Newton’s method is used as an outer nonlinear iteration and for the inner linear sub problems, the direct solver UMFPACK is utilized. The results have been compared with the reference data present in the literature and a good agreement is found.

A new family of nonconforming finite elements on quadrilaterals

In this paper, we generalize one first order nonconforming quadrilateral finite element proposed by Lin, Tobiska and Zhou to any odd order. We construct degrees of freedom for shape function spaces for this family of elements and show their unisolvency. In addition, we present a medium a priori error analysis for this family of nonconforming elements on general quadrilateral meshes. Compared with the classical error analysis of the nonconforming finite element method, the a priori analysis herein only needs the H1 regularity of the exact solution. Numerics are presented to demonstrate theoretical results which in particular show dependence of convergence on mesh distortion parameters α.

Mixed finite elements for elasticity on quadrilateral meshes

We present stable mixed finite elements for planar linear elasticity on general quadrilateral meshes. The symmetry of the stress tensor is imposed weakly and so there are three primary variables, the stress tensor, the displacement vector field, and the scalar rotation. We develop and analyze a stable family of methods, indexed by an integer r ? 2 and with rate of convergence in the L2 norm of order r for all the variables. The methods use Raviart---Thomas elements for the stress, piecewise tensor product polynomials for the displacement, and piecewise polynomials for the rotation. We also present a simple first order element, not belonging to this family. It uses the lowest order BDM elements for the stress, and piecewise constants for the displacement and rotation, and achieves first order convergence for all three variables.

Constraint-Free Adaptive FEMs on Quadrilateral Nonconforming Meshes

Finite element methods (FEMs) on nonconforming meshes have been much studied in the literature. In all earlier works on such methods , some constraints must be imposed on the degrees of freedom on the edge/face with hanging nodes in order to maintain continuity, which make the numerical implementation more complicated. In this paper, we present two FEMs on quadrilateral nonconforming meshes which are constraint-free. Furthermore, we establish the corresponding residual-based a posteriori error reliability and efficiency estimation for general quadrilateral meshes. We also present extensive numerical testing results to systematically compare the performance among three adaptive quadrilateral FEMs: the constraint-free adaptive $$\mathbb Q _1$$ Q 1 FEM on quadrilateral nonconforming meshes with hanging nodes developed herein, the adaptive $$\mathbb Q _1$$ Q 1 FEM based on quadrilateral red-green refinement without any hanging node recently proposed in Zhao et al. (SIAM J Sci Comput 3(4):2099---2120, 2010), and the classical adaptive $$\mathbb Q _1$$ Q 1 FEM on quadrilateral nonconforming meshes with constraints on hanging nodes. Some extensions are also included in this paper.

Some remarks on quadrilateral mixed finite elements

It is well known that quadrilateral finite elements may produce unsatisfactory results when used on distorted meshes. It turns out that many commonly used finite elements achieve suboptimal convergence properties on distorted quadrilaterals; among such elements we recall in particular serendipity (trunk) scalar elements and basically all vectorial elements for the approximation of problems involving the functional space H ( div ) (like Raviart–Thomas or Brezzi–Douglas–Marini spaces for Darcy flow). In two space dimensions, a similar remark applies to edge finite elements for the approximation of Maxwell’s problems involving the space H ( curl ) . On the other hand, mimetic finite differences have become popular for the approximation of problems involving H ( div ) on very general geometries. The aim of this paper is to show how to use the ideas of mimetic finite differences for the stabilization of Raviart–Thomas element on general quadrilateral meshes. It turns out that such stabilization can be performed by a slight modification of the standard Raviart–Thomas element which does not increase significantly the computational cost of the original scheme.

Residual Distribution Schemes on Quadrilateral Meshes

We propose an investigation of the residual distribution schemes for the numerical approximation of two-dimensional hyperbolic systems of conservation laws on general quadrilateral meshes. In comparison to the use of triangular cells, usual basic features are recovered, an extension of the upwinding concept is given, and a Lax–Wendroff type theorem is adapted for consistency. We show how to retrieve many variants of standard first and second-order accurate schemes. They are proven to satisfy this theorem. An important part of this paper is devoted to the validation of these schemes by various numerical tests for scalar equations and the Euler equations system for compressible fluid dynamics on non Cartesian grids. In particular, second-order accuracy is reached by an adaptation of the Linearity preserving property to quadrangle meshes. We discuss several choices as well as the convergence of iterative method to steady state. We also provide examples of schemes that are not constructed from an upwinding principle

EDGE ELEMENT COMPUTATION OF MAXWELL'S EIGENVALUES ON GENERAL QUADRILATERAL MESHES

Recent results prove that Nédélec edge elements do not achieve optimal rate of approximation on general quadrilateral meshes. In particular, lowest order edge elements provide stable but non-convergent approximation of Maxwell's eigenvalues. In this paper we analyze a modification of standard edge element that restores the optimality of the convergence. This modification is based on a projection technique that can be interpreted as a reduced integration procedure.

Eigenmodes computation on quadrilateral meshes

We approximate a fluid-structure interaction eigenvalue problem by means of finite elements on quadrilateral meshes. The problem under consideration looks simple and has been the object of a wide investigation: actually, no standard numerical scheme achieves acceptable results in presence of field singularities. We consider a three-field mixed method introduced by Bathe et al. and based on rectangular meshes (see [3, 12]). Recent results show that particular care has to be taken into account when dealing with finite element approximation on general quadrilateral meshes. We present numerical experiments on several sequences of meshes which definitely show how the “local” approach has to be discarded, while the “global” approach gives reasonable results.

Conservative rezoning (remapping) for general quadrilateral meshes

A classic problem in Lagrangian numerical hydrodynamics is the conservative transfer of quantities from an old, distorted mesh to a new mesh. The same problem arises whenever the mesh is changed as, for example, in adaptive mesh techniques. This transfer of information is an interpolation process which is frequently called rezoning (or remapping). The general problem of conservative rezoning from one arbitrary mesh to another may be formulated as follows: m k = ∫∫∫ V k ϱ( r) dV. That is, we compute the mass m k of each cell of the new mesh by integrating the known density distribution in the old mesh over the cell volume V k . A direct intregration is generally prohibitive. We show, however, that it is possible to convert this integral to a surface integral by the appropriate use of the divergence theorem, thus greatly reducing the complexity of the problem. For two-dimensional general quadrilateral meshes the resulting method is exact and particularly simple.

Studies in Bayesian Econometrics and Statistics, In honor of Leonard J. Savage

We construct a finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes, where the strict convexity restriction on the meshes is removed. The main contributions of this paper include three aspects. Firstly, the classical finite volume element (C-FVE) method is extended to severely distorted quadrilateral meshes even with concave cells by virtue of a new overlapping dual partition and a new gradient approximation. In fact, the choice of this dual partition and gradient approximation is also conducive to the construction of a monotone scheme. Secondly, a new monotonicity correction is suggested, based on which we obtain a monotone finite volume element (M-FVE) method. The resulting M-FVE method still keeps the local conservation and is easy for implementation. Finally, we analyze theoretically the truncation error and the monotonicity for this scheme. Besides, the existence of a solution to this nonlinear scheme is proved by applying the Brouwer's fixed point theorem. Numerical results demonstrate that the M-FVE method has the approximate second-order accuracy and preserves well the positivity of the solution for both isotropic and anisotropic diffusion problems on severely distorted quadrilateral meshes.