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Abstract 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 (ABF k , 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 h k + 1 , where h is the mesh diameter, in the H (div) -norm. We show that displacement and rotation are also approximated to order h k + 1 in the L 2 -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. Highlights • New stable finite element spaces for the mixed approximation of planar linear elasticity on quadrilaterals. • O (h k + 1) -approximation of the stress field in the H (div) -norm on meshes of nondegenerate, convex quadrilaterals, with k ≥ 1 (existing elements do not have this property). • Special treatment for the lowest order case (k = 0) with improved convergence rates for stress and rotation, in the L 2 -norm. • The formulation is suitable for both compressible and incompressible isotropic problems. [ABSTRACT FROM AUTHOR]

The coupling of free fluid and porous medium flows, governed by the Stokes and Darcy systems in their mixed velocity–pressure formulations, is addressed in two space dimensions. A stabilized mixed hybrid finite element method is proposed using discontinuous finite element spaces for the velocity and pressure approximations. Lagrange multipliers, associated with the traces of the velocity fields, are introduced to weakly impose the transmission conditions on the edges of the elements in both domains. The Beavers–Joseph–Saffman interface conditions on the interface between Stokes and Darcy domains also are naturally imposed by the same Lagrange multipliers. By condensing the velocity and pressure degrees-of-freedom at the element level the global system is assembled involving only the degrees-of-freedom associated with the multipliers. Stability of the proposed formulation is proved for ample choices of the finite element spaces including equal order of approximations for all fields on triangle and quadrilateral elements. Numerical experiments are performed illustrating flexibility, robustness and optimal rates of convergence. [ABSTRACT FROM AUTHOR]

Primal stabilized hybrid finite element methods for the linear elasticity problem are proposed consisting of locally discontinuous Galerkin problems in the primal variable coupled to a global problem in the multiplier which is identified with the trace of the displacement field. Numerical analysis, covering both continuous or discontinuous interpolations of the multiplier, shows that the proposed formulation preserves the main properties of the associate DG method such as consistency, stability, boundedness and optimal rates of convergence in the energy norm, and in the L2(Ω) norm for adjoint consistent formulations. Convergence studies confirm the optimal rates of convergence predicted by the numerical analysis presented here and a local post-processing technique is proposed to recover stress approximations with improved rates of convergence in H(div) norm. [ABSTRACT FROM AUTHOR]

Abstract: A discontinuous finite element formulation is presented for Helmholtz equation. Continuity is relaxed locally in the interior of the element instead of across the element edges. The interior shape functions can be viewed as discontinuous bubbles and the corresponding degrees of freedom can be eliminated at element level by static condensation yielding a global matrix topologically equivalent to those of classical C 0 finite element approximations. The discontinuous bubbles are introduced to consistently add stability to the formulation, preserving the optimal rates of convergence and improving accuracy. A crucial point of the discontinuous formulation relies on the choice of the stabilization parameters related to the weak enforcement of continuity inside each element. Explicit values of these stabilization parameters minimizing the pollution effect are presented for uniform meshes. The accuracy and stability of the proposed formulation for bilinear shape functions are demonstrated in several numerical examples. [Copyright &y& Elsevier]

A local residual error estimator based on the least-square residuals of the balance equation, constitutive relation and irrotationality condition is presented to enable adaptive finite element analyses of Poisson’s problems. The efficiency of the residual error indicator and the quality of the L2 error indicator are investigated in problems with sharp and relatively sharp concentration of gradients. Both error estimators are highly dependent on the accuracy of the recovered or post-processed derivatives. Two types of post-processing are adopted: the L2 projection and a local macroelement technique. In the proposed macroelement post-processing technique, the derivatives are recovered by solving local variational problems involving the residuals of the balance equation irrotationality condition and constitutive relation at special superconvergence points. [Copyright &y& Elsevier]

To get the stationary patterns in the approximation of reaction diffusion systems, the computation of evolution equations on large intervals of time is required. In this paper, a new nonlinear Galerkin method based on finite element discretization is presented for the approximation of reaction diffusion equations on large intervals. The new scheme is based on two different finite element spaces defined respectively on one subspace of lower-degree shape function and one of higher-degree shape function. Nonlinearity and time dependence are both treated on the lower-degree space and only a fixed stationary equation needs to be solved on the higher-degree space at each time level. The proof of the stability and convergence of the method is presented in one dimension. Numerical solutions of some well-studied reaction–diffusion systems are presented to demonstrate the effectiveness of the nonlinear Galerkin method. [ABSTRACT FROM AUTHOR]

Abstract: A stabilized mixed discontinuous Galerkin (SMDG) method based on Brezzi–Hughes–Marini–Masud [F. Brezzi, T.J.R. Hughes, L.D. Marini, A. Masud, Mixed discontnuous Galerkin methods for Darcy flow, J. Sci. Comput. 22 (2005) 119–145.] is proposed to solve a thermally coupled nonlinear elliptic system modeling a large class of engineering problems. A fixed point algorithm is adopted to solve the nonlinear systems. Convergence analysis and error estimates are presented for equal order linear or bilinear discontinuous Lagrangian finite element interpolations for all fields. Numerical results are presented confirming the predicted convergence rates and illustrating the performance of the proposed formulation solving problems with globally stable and blowing up solutions. [Copyright &y& Elsevier]

A new combined method is constructed for solving incompressible miscible displacement in porous media. In this procedure, a hybrid mixed element method is constructed for the pressure and velocity equation, while a symmetric discontinuous Galerkin finite element method is proposed for the concentration equation. The introduction of these two numerical methods not only makes the coefficient matrixes symmetric positive definite, but also conserves the local mass balance. The stability and consistency of the method are analyzed and the optimal error estimate in L ∞ ( L 2 ) for velocity and concentration and the super convergence in L ∞ ( H 1 ) for pressure are derived. Finally, some numerical results are presented. [ABSTRACT FROM AUTHOR]