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This paper is devoted to the construction and analysis of the finite element approximations for the H(D) convection-diffusion problems, where D can be chosen as grad, curl or div in 3D case. An essential feature of these constructions is to properly average the PDE coefficients on the sub-simplexes. The schemes are of the class of exponential fitting methods that result in special upwind schemes when the diffusion coefficient approaches to zero. Their well-posedness are established for sufficiently small mesh size assuming that the convection-diffusion problems are uniquely solvable. Convergence of first order is derived under minimal smoothness of the solution. Some numerical examples are given to demonstrate the robustness and effectiveness for general convection-diffusion problems.The theory and numerical analysis of such a scalar convection-diffusion equation are well-studied in the literature. It is well known that, for small α, boundary layers may appear in the solution of (1.2) and standard finite element methods may suffer from strong numerical oscillations and instabilities if the mesh size is not small enough.Numerous studies on the stable discretization of scalar convection-diffusion have been published. In the finite element methods, various special strategies have been developed, including the stabilized discontinuous Galerkin method [33,10], SUPG method [12,22,14], bubble function stabilization [11,8,7,23], local projection stabilization [25], edge stabilization and continuous interior penalty method [16,13,15]. Other studies do not require the characteristics to be specified, such as exponential fitting [9,38,19,34,4] and Petrov-Galerkin method [35,18].The H(curl) and H(div) convection-diffusion problems have received more and more attention from numerical computation. The discretization of the general convection, known as extrusion, has been discussed via Whitney forms in [6]. For the pure advection problem, the stabilized Galerkin method has been extended from 0form [10] to 1-form [30] and k-form [28,32]. These discretizations of the advection problem, along with the proper discretization of the diffusion term, are feasible to tackle the general convection-diffusion problems. Besides the Eulerian method, the semi-Lagrangian method can be applied to the time-dependent convection-diffusion problems for differential forms [28,31,29].More specifically, we are motivated by the Edge-Averaged Finite Element (EAFE) method for scalar convection-diffusion problem proposed by Xu and Zikatanov [38]. There are two main advantages to using EAFE: (1) The monotonicity of EAFE can be established for a very general class of meshes; (2) The local stiffness matrix of EAFE can easily be obtained by modifying that of standard Poisson. A construction that ensures the general SPD diffusion coefficient matrix was proposed in [34]. A high-order Scharfetter-Gummel scheme, known as a high-order extension of EAFE, was given in [4]. Similar to (1.2), the standard finite element methods also seriously suffer from numerical ins...

In this work, we propose a family of interior penalty nonconforming finite element methods for 2m-th order partial differential equations in R n , for any m ≥ 0, n ≥ 1. For the nonconforming finite element, the shape function space consists of all polynomials with a degree not greater than m and is hence minimal. This family of finite element spaces has some natural inclusion properties as in the corresponding Sobolev spaces in the continuous cases. By applying the interior penalty to the bilinear form, we establish quasi-optimal error estimates in the energy norm, provided that the corresponding conforming relatives exist. These theoretical results are further validated by the numerical tests.

In this paper, we present a family of new mixed finite element methods for linear elasticity for both spatial dimensions n = 2, 3, which yields a conforming and strongly symmetric approximation for stress. Applying P k+1 − P k as the local approximation for the stress and displacement, the mixed methods achieve the optimal order of convergence for both the stress and displacement when k ≥ n. For the lower order case (n − 2 ≤ k < n), the stability and convergence still hold on some special grids. The proposed mixed methods are efficiently implemented by hybridization, which imposes the inter-element normal continuity of the stress by a Lagrange multiplier. Then, we develop and analyze multilevel solvers for the Schur complement of the hybridized system in the two dimensional case. Provided that no nearly singular vertex on the grids, the proposed solvers are proved to be uniformly convergent with respect to both the grid size and Poisson's ratio. Numerical experiments are provided to validate our theoretical results.