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## Theory and Application of Characteristic Finite Element Domain Decomposition Procedures for Coupled System of Dynamics of Fluids in Porous Media

### Acta Mathematicae Applicatae Sinica, English Series (2007-04-01) 23: 255-268 , April 01, 2007

###
*Abstract*

For a coupled system of multiplayer dynamics of fluids in porous media, the characteristic finite element domain decomposition procedures applicable to parallel arithmetic are put forward. Techniques such as calculus of variations, domain decomposition, characteristic method, negative norm estimate, energy method and the theory of prior estimates are adopted. Optimal order estimates in L^{2} norm are derived for the error in the approximate solution.

## Isoparametric C 0 interior penalty methods for plate bending problems on smooth domains

### Calcolo (2013-03-01) 50: 35-67 , March 01, 2013

In this paper we develop isoparametric *C*^{0} interior penalty methods for plate bending problems on smooth domains. The orders of convergence of these methods are shown to be optimal in the energy norm. We also consider the convergence of these methods in lower order Sobolev norms and discuss subparametric *C*^{0} interior penalty methods. Numerical results that illustrate the performance of these methods are presented.

## Two-grid partition of unity method for second order elliptic problems

### Applied Mathematics and Mechanics (2008-04-01) 29: 527-533 , April 01, 2008

A two-grid partition of unity method for second order elliptic problems is proposed and analyzed. The standard two-grid method is a local and parallel method usually leading to a discontinuous solution in the entire computational domain. Partition of unity method is employed to glue all the local solutions together to get the global continuous one, which is optimal in *H*^{1}-norm. Furthermore, it is shown that the *L*^{2} error can be improved by using the coarse grid correction. Numerical experiments are reported to support the theoretical results.

## Interpolation error estimates for mean value coordinates over convex polygons

### Advances in Computational Mathematics (2013-08-01) 39: 327-347 , August 01, 2013

In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in Gillette et al. (Adv Comput Math 37(3), 417–439, 2012), we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying certain simple geometric restrictions. This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradients of the mean value coordinates do not become large as interior angles of the polygon approach *π*.

## An adaptive residual local projection finite element method for the Navier–Stokes equations

### Advances in Computational Mathematics (2014-12-01) 40: 1093-1119 , December 01, 2014

This work proposes and analyses an adaptive finite element scheme for the fully non-linear incompressible Navier-Stokes equations. A residual a posteriori error estimator is shown to be effective and reliable. The error estimator relies on a Residual Local Projection (RELP) finite element method for which we prove well-posedness under mild conditions. Several well-established numerical tests assess the theoretical results.

## An efficient and reliable residual-type a posteriori error estimator for the Signorini problem

### Numerische Mathematik (2015-05-01) 130: 151-197 , May 01, 2015

We derive a new a posteriori error estimator for the Signorini problem. It generalizes the standard residual-type estimators for unconstrained problems in linear elasticity by additional terms at the contact boundary addressing the non-linearity. Remarkably these additional contact-related terms vanish in the case of so-called full-contact. We prove reliability and efficiency for two- and three-dimensional simplicial meshes. Moreover, we address the case of non-discrete gap functions. Numerical tests for different obstacles and starting grids illustrate the good performance of the a posteriori error estimator in the two- and three-dimensional case, for simplicial as well as for unstructured mixed meshes.

## Anisotropic nonconforming Crouzeix-Raviart type FEM for second-order elliptic problems

### Applied Mathematics and Mechanics (2012-02-01) 33: 243-252 , February 01, 2012

The nonconforming Crouzeix-Raviart type linear triangular finite element approximate to second-order elliptic problems is studied on anisotropic general triangular meshes in 2D satisfying the maximal angle condition and the coordinate system condition. The optimal-order error estimates of the broken energy norm and *L*^{2}-norm are obtained.

## Back Matter - hp-Finite Element Methods for Singular Perturbations

### hp-Finite Element Methods for Singular Perturbations (2002-01-01): 1796 , January 01, 2002

## Robust a Posteriori Error Estimates for Conforming Discretizations of Diffusion Problems with Discontinuous Coefficients on Anisotropic Meshes

### Journal of Scientific Computing (2015-08-01) 64: 368-400 , August 01, 2015

In this paper, we study a posteriori estimates for different numerical methods of diffusion problems with discontinuous coefficients on anisotropic meshes, in particular, which can be applied to vertex-centered and cell-centered finite volume, finite difference and piecewise linear finite element methods. Based on the stretching ratios of the mesh elements, we improve a posteriori estimates developed by Vohralík (J Sci Comput 46:397–438, 2011), which are reliable and efficient on isotropic meshes but fail on anisotropic ones (see the numerical results of the paper). Without the assumption that the meshes are shape-regular, the resulting mesh-dependent error estimators are shown to be reliable and efficient with respect to the error measured either as the energy norm of the difference between the exact and approximate solutions, or as a dual norm of the residual, as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution. In other words, they are equivalent to the estimates of Vohralík in the case of isotropic meshes and proved to be robust on anisotropic meshes as well. Based on $$\mathbf{H}(\mathrm {div})$$ -conforming, locally conservative flux reconstruction, we suggest two different constructions of the equilibrated flux with the anisotropy of mesh, which is essential to the robustness of our estimates on anisotropic meshes. Numerical experiments in 2D confirm that our estimates are reliable and efficient on anisotropic meshes.

## Efficient solution of nonlinear elliptic problems using hierarchical matrices with Broyden updates

### Computing (2007-12-01) 81: 239-257 , December 01, 2007

### Summary

The numerical solution of nonlinear problems is usually connected with Newton’s method. Due to its computational cost, variants (so-called inexact and quasi–Newton methods) have been developed in which the arising inverse of the Jacobian is replaced by an approximation. In this article we present a new approach which is based on Broyden updates. This method does not require to store the update history since the updates are explicitly added to the matrix. In addition to updating the inverse we introduce a method which constructs updates of the *LU* decomposition. To this end, we present an algorithm for the efficient multiplication of hierarchical and semi-separable matrices. Since an approximate *LU* decomposition of finite element stiffness matrices can be efficiently computed in the set of hierarchical matrices, the complexity of the proposed method scales almost linearly. Numerical examples demonstrate the effectiveness of this new approach.