This is the first in a series of papers in which a new gradient recovery method is introduced and analyzed. It is proved that the method is superconvergent for translation invariant finite element spaces of any order. The method maintains the simplicity, efficiency, and superconvergence properties of the Zienkiewicz--Zhu patch recovery method. In addition, for uniform triangular meshes, the method is superconvergent for the linear element under the chevron pattern, and ultraconvergent at element edge centers for the quadratic element under the regular pattern. Applications of this new gradient recovery technique will be discussed in forthcoming papers.

A New Finite Element Gradient Recovery Method: Superconvergence Property

Some recovery type error estimators for linear finite elements are analyzed under O(h 1+α ) (α > 0) regular grids. Superconvergence of order O(h 1+ρ ) (0 < p < α) is established for recovered gradients by three different methods. As a consequence, a posteriori error estimators based on those recovery methods are asymptotically exact.

/pdf/analysis-of-recovery-type-a-posteriori-error-estimators-for-1uxdwde2kl.pdf

Analysis of recovery type a posteriori error estimators for mildly structured grids

We present a new, naturally parallelizable, accurate numerical method for the solution of transport-dominated diffusion processes in heterogeneous porous media For the discretization in time of one of the governing partial differential equations, we introduce a new characteristics-based procedure which is mass conservative, the modified method of characteristics with adjusted advection (MMOCAA) Hybridized mixed finite elements are used for the spatial discretization of the equations and a new strip-based domain decomposition procedure is applied towards the solution of the resulting algebraic problems We consider as a model problem the two-phase immiscible displacement in petroleum reservoirs A very detailed description of the numerical method is presented Following that, numerical experiments are presented illustrating the important features of the new method and comparing computed results with ones derived from previous, related techniques

On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs

The equilibrated residual method for a posteriori error estimation is extended to nonconforming finite element schemes for the approximation of linear second order elliptic equations where the permeability coefficient is allowed to undergo large jumps in value across interfaces between differing media. The estimator is shown to provide a computable upper bound on the error and, up to a constant depending only on the geometry, provides two-sided bounds on the error. The robustness of the estimator is also studied and the dependence of the constant on the jumps in permeability is given explicitly.

/pdf/robust-a-posteriori-error-estimation-for-nonconforming-19ajy3938d.pdf

Robust A Posteriori Error Estimation for Nonconforming Finite Element Approximation

We propose several formulations for recovering discontinuous coefficients in elliptic problems by using total variation (TV) regularization. The motivation for using TV is its well-established ability to recover sharp discontinuities. We employ an augmented Lagrangian variational formulation for solving the output-least-squares inverse problem. In addition to the basic output-least-squares formulation, we introduce two new techniques for handling large observation errors. First, we use a filtering step to remove as much of the observation error as possible. Second, we introduce two extensions of the output-least-squares model; one model employs observations of the gradient of the state variable while the other uses the flux. Numerical experiments indicate that the combination of these two techniques enables us to successfully recover discontinuous coefficients even under large observation errors.

/pdf/identification-of-discontinuous-coefficients-in-elliptic-57y59re3y2.pdf

Identification of Discontinuous Coefficients in Elliptic Problems Using Total Variation Regularization

Coupling Mortar Finite Element and Boundary Element Methods for 2D Navier-Stokes Equations Courant Element: Before and After Iterative Methods for Solving Stiff Elliptic Problems Straight and Curved Finite Elements of Class C1 and Some Applications to Thin Shell Problems Exact Controllability to Solve the Helmholtz Equation with Absorbing Boundary Conditions Cubic Version of FEM in Elliptic Problems with Interfaces and Singularities Least Squares Mixed Finite Elements Turbulence Modelling in Finite Element Industrial Applications Necessary and Sufficient Conditions for the Numerical Approximation of a Partial Differential Equation Depending on a Small Parameter Efficient Solution Methods for Compressible Flow Computations Parallel Finite Volume Algorithms for Solving the Time-Domain Maxwell Equations on Nonstructured Meshes Coupling Between Nonlinear Maxwell and Heat Equations for an Induction Heating Problem: Modelling and Numerical Methods Solving the 3D Harmonic Maxwell Equations with Finite Elements, Lagrange Multipliers, and Iterative Methods Some Applications of the Hierarchic High Order MITC Finite Elements for Reissner-Mindlin Plates Domain Decomposition for Immiscible Displacement in Single Porosity Systems An Error Estimator for Nonconforming Approximations of a Nonlinear Problem Some Observations on Raviart-Thomas Mixed Finite Elements in p Extension for Parabolic Problems Mixed Finite Element Methods in Fluid Structure Systems A Black-Box Solver for the Solution of General Nonlinear Functional Equations by Mixed FEM A Remark on the Asymptotic Behaviour of Parabolic Variational Inequalities and Their Finite Element Approximation by the Courant Element Domain Decomposition vs. Adaptivity Material Optimization of Composites. Part Contents.

Finite element methods : fifty years of the Courant element

A black-box solver for the solution of nonlinear functional equations is a compromise of exibilty, eeciency and stability of the algorithms. We will present a general form of functional equation, a general error indicator and the eecient interaction of the error indicator and the Newton-Raphson iteration during the solution of the problem arising from the discretization by mixed nite element methods.

A Black-Box Solver for the Solution of General Nonlinear Functional Equations by Mixed FEM

A Posteriori Error Analysis in the Finite Element Method

Necessary and Sufficient Conditions for the Numerical Approximation of a Partial Differential Equation Depending on a Small Parameter

A Wavelet Transform Based on Periodic Splines and Finite Element Method

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