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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: We propose and analyze a stabilized hybrid finite element method for elliptic problems consisting of locally discontinuous Galerkin problems in the primal variable coupled to a globally continuous problem in the multiplier. Numerical analysis 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 norm for adjoint consistent formulations. For using an element based data structure, it has basically the same complexity and computational cost of classical conforming finite element methods. Convergence studies confirm the optimal rates of convergence predicted by the numerical analysis presented here, with accuracy equivalent or even better than the corresponding DG approximations. [Copyright &y& Elsevier]

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]