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We develop a new numerical method, namely, a locking-free staggered cell-centered discontinuous Galerkin method for linear elasticity on fairly general meshes. The method is well suited for general...

In this paper, we introduce staggered discontinuous Galerkin methods for the stationary Stokes flow on polygonal meshes. The proposed method is based on the pseudostress-velocity formulation. A Lag...

In this paper, we first propose and analyze a locally conservative, lowest order staggered discontinuous Galerkin method of minimal dimension on general quadrilateral/polygonal meshes for elliptic ...

The first-order div least squares finite element methods (LSFEMs) allow for an immediate a posteriori error control by the computable residual of the least squares functional. This paper establishes an adaptive refinement strategy based on some equivalent refinement indicators. Since the first-order div LSFEM measures the flux errors in $H$(div), the data resolution error measures the $L^2$ norm of the right-hand side $f$ minus the piecewise polynomial approximation $\Pi f$ without a mesh-size factor. Hence the data resolution term is neither an oscillation nor of higher order and consequently requires a particular treatment, e.g., by the thresholding second algorithm due to Binev and DeVore. The resulting novel adaptive LSFEM with separate marking converges with optimal rates relative to the notion of a nonlinear approximation class.

The stress-velocity formulation of the stationary Stokes problem allows an Arnold--Winther mixed finite element formulation with some superconvergent reconstruction of the velocity. This local postprocessing gives rise to two reliable a posteriori error estimators which recover optimal convergence order for the stress error estimates. The theoretical results are investigated in numerical benchmark examples.

The pseudostress-velocity formulation of the stationary Stokes problem allows a Raviart-Thomas mixed finite element formulation with quasi-optimal convergence and some superconvergent reconstruction of the velocity. This local postprocessing gives rise to some averaging a posteriori error estimator with explicit constants for reliable error control. Standard residual-based explicit a posteriori error estimation is shown to be reliable and efficient and motivates adaptive mesh-refining algorithms. Numerical experiments confirm our theoretical findings and illustrate the accuracy of the guaranteed upper error bounds even with reduced regularity.

We study the mixed finite element approximation of the second order elliptic problem with gradient nonlinearities in two and three space dimensions. Existence and uniqueness of the approximate solution are proved and optimal order a priori error estimates in $L^m(\Omega)$ are obtained. Also, reliable and efficient a posteriori error estimators measured in the $L^m(\Omega)$-norm are derived. Numerical examples are provided to illustrate the performance of the proposed estimator.

A new family of hybrid discontinuous Galerkin methods is studied for second-order elliptic equations. Our proposed method is a generalization of the cell boundary element (CBE) method [Y. Jeon and E.-J. Park, Appl. Numer. Math., 58 (2008), pp. 800-814], which allows high order polynomial approximations. Our method can be viewed as a hybridizable discontinuous Galerkin method [B. Cockburn, J. Gopalakrishnan, and R. Lazarov, SIAM J. Numer. Anal., 47 (2009), pp. 1319-1365] using a Bauman-Oden-type local solver. The method conserves the mass in each element and the average flux is continuous across the interelement boundary for even-degree polynomial approximations. Optimal order error estimates measured in the energy norm are proved. Numerical examples are presented to show the performance of the method.