A FEM–DtN formulation for a non‐linear exterior problem in incompressible elasticity

In this paper, we combine the usual finite element method with a Dirichlet‐to‐Neumann (DtN) mapping, derived in terms of an infinite Fourier series, to study the solvability and Galerkin approximations of an exterior transmission problem arising in non‐linear incompressible 2d‐elasticity. We show that the variational formulation can be written in a Stokes‐type mixed form with a linear constraint and a non‐linear main operator. Then, we provide the uniqueness of solution for the continuous and discrete formulations, and derive a Cea‐type estimate for the associated error. In particular, our error analysis considers the practical case in which the DtN mapping is approximated by the corresponding finite Fourier series. Finally, a reliable a posteriori error estimate, well suited for adaptive computations, is also given. Copyright © 2003 John Wiley & Sons, Ltd.     

Finite‐time synchronization of a class of N‐coupled complex partial differential systems with time‐varying delay

This study examines finite‐time synchronization for a class of N‐coupled complex partial differential systems (PDSs) with time‐varying delay. The problem of finite‐time synchronization for coupled drive‐response PDSs with time‐varying delay is similarly considered. The synchronization error dynamic of the PDSs is defined in the q‐dimensional spatial domain. We construct a feedback controller to achieve finite‐time synchronization. Sufficient conditions are derived by using the Lyapunov‐Krasoviskii stability approach and inequalities technology to ensure that the proposed networks achieve synchronization in finite time. The proposed systems demonstrate extensive application. Finally, an example is used to verify the theoretical results.     

Robust residual‐ and recovery‐based a posteriori error estimators for a multipoint flux mixed finite element discretization of interface problems

We consider a posteriori error estimation for a multipoint flux mixed finite element method for two‐dimensional elliptic interface problems. Within the class of modified quasi‐monotonically distributed coefficients, we derive a residual‐type a posteriori error estimator of the weighted sum of the scalar and flux errors which is robust with respect to the jumps of the coefficients. Moreover, we develop robust implicit and explicit recovery‐type estimators through gradient recovery in an H(curl)‐conforming finite element space. In particular, we apply a modified L² projection in the implicit recovery procedure so as to reduce the computational cost of the recovered gradient. Numerical experiments confirm the theoretical results.     

Guaranteed velocity error control for the pseudostress approximation of the Stokes equations

The pseudostress approximation of the Stokes equations rewrites the stationary Stokes equations with pure (but possibly inhomogeneous) Dirichlet boundary conditions as another (equivalent) mixed scheme based on a stress in H(div) and the velocity in L². Any standard mixed finite element function space can be utilized for this mixed formulation, e.g., the Raviart‐Thomas discretization which is related to the Crouzeix‐Raviart nonconforming finite element scheme in the lowest‐order case. The effective and guaranteed a posteriori error control for this nonconforming velocity‐oriented discretization can be generalized to the error control of some piecewise quadratic velocity approximation that is related to the discrete pseudostress. The analysis allows for local inf‐sup constants which can be chosen in a global partition to improve the estimation. Numerical examples provide strong evidence for an effective and guaranteed error control with very small overestimation factors even for domains with large anisotropy.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1411–1432, 2016     

On the Convergence of Adaptive Finite Element Methods

State of the art simulations in computational mechanics aim reliability and efficiency via adaptive finite element methods (AFEMs) with a posteriori error control. The a priori convergence of finite element methods is justified by the density property of the sequence of finite element spaces which essentially assumes a quasi‐uniform mesh‐refining. The advantage is guaranteed convergence for a large class of data and solutions; the disadvantage is a global mesh refinement everywhere accompanied by large computational costs. AFEMs automatically refine exclusively wherever the refinement indication suggests to do so and so violate the density property on purpose. Then, the a priori convergence of AFEMs is not guaranteed automatically and, in fact, crucially depends on algorithmic details. The advantage of AFEMs is a more effective mesh accompanied by smaller computational costs in many practical examples; the disadvantage is that the desirable error reduction property is not always guaranteed a priori. Efficient error estimators can justify a numerical approximation a posteriori and so achieve reliability. But it is not clear from the start that the adaptive mesh‐refinement will generate an accurate solution at all. This paper discusses particular versions of an AFEMs and their analyses for error reduction, energy reduction, and convergence results for linear and nonlinear problems. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A posteriori error estimates in quantities of interest for the finite element heterogeneous multiscale method

We present an “a posteriori” error analysis in quantities of interest for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. The multiscale method is based on a macro‐to‐micro formulation, where the macroscopic physical problem is discretized in a macroscopic finite element space, and the missing macroscopic data are recovered on‐the‐fly using the solutions of corresponding microscopic problems. We propose a new framework that allows to follow the concept of the (single‐scale) dual‐weighted residual method at the macroscopic level in order to derive a posteriori error estimates in quantities of interests for multiscale problems. Local error indicators, derived in the macroscopic domain, can be used for adaptive goal‐oriented mesh refinement. These error indicators rely only on available macroscopic and microscopic solutions. We further provide a detailed analysis of the data approximation error, including the quadrature errors. Numerical experiments confirm the efficiency of the adaptive method and the effectivity of our error estimates in the quantities of interest. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Uniform methods for semilinear problems with attractive boundary turning points

An upwind finite‐difference scheme for the numerical solution of semilinear convection‐diffusion problems with attractive boundary turning points is considered. We show that the maximum nodal error is bounded by a special weighted ℓ₁‐type norm of the truncation error. This result is used to establish uniform convergence with respect to the perturbation parameter on Shishkin meshes.     

An adaptive finite element method for a time‐dependent Stokes problem

In this article, we conduct an a posteriori error analysis of the two‐dimensional time‐dependent Stokes problem with homogeneous Dirichlet boundary conditions, which can be extended to mixed boundary conditions. We present a full time–space discretization using the discontinuous Galerkin method with polynomials of any degree in time and the ? ₂ ? ?₁ Taylor–Hood finite elements in space, and propose an a posteriori residual‐type error estimator. The upper bounds involve residuals, which are global in space and local in time, and an L ²‐error term evaluated on the left‐end point of time step. From the error estimate, we compute local error indicators to develop an adaptive space/time mesh refinement strategy. Numerical experiments verify our theoretical results and the proposed adaptive strategy.     

Finite element method for two‐dimensional time‐fractional tricomi‐type equations

In this article, we consider the finite element method (FEM) for two‐dimensional linear time‐fractional Tricomi‐type equations, which is obtained from the standard two‐dimensional linear Tricomi‐type equation by replacing the first‐order time derivative with a fractional derivative (of order α, with 1 <α< 2 ). The method is based on finite element method for space and finite difference method for time. We prove that the method is unconditionally stable, and the error estimate is presented. The comparison of the FEM results with the exact solutions is made, and numerical experiments reveal that the FEM is very effective. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

A posteriori error estimators for optimal distributed control governed by the first‐order linear hyperbolic equation: DG method

In this article, We analyze the ‐version of the discontinuous Galerkin finite element method (DGFEM) for the distributed first‐order linear hyperbolic optimal control problems. We derive a posteriori error estimators on general finite element meshes which are sharp in the mesh‐width . These error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems. For the DGFEM we admit very general irregular meshes. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Nitsche type mortaring for elliptic problems with corner singularities

The paper deals with Nitsche type mortaring as a finite element method (FEM) for treating non‐matching meshes of triangles at the interface of some domain decomposition. The approach is applied to the Poisson equation with Dirichlet conditions for the case that the interface passes re‐entrant corners of the domain and local mesh refinement is applied. Some properties of the finite element scheme and error estimates in a discrete H¹‐like and in the L₂‐norm are proved.     

Nonconforming finite element methods with subgrid viscosity applied to advection‐diffusion‐reaction equations

A nonconforming (Crouzeix–Raviart) finite element method with subgrid viscosity is analyzed to approximate advection‐diffusion‐reaction equations. The error estimates are quasi‐optimal in the sense that keeping the Péclet number fixed, the estimates are suboptimal of order in the mesh size for the L²‐norm and optimal for the advective derivative on quasi‐uniform meshes. The method is also reformulated as a finite volume box scheme providing a reconstruction formula for the diffusive flux with local conservation properties. Numerical results are presented to illustrate the error analysis. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Finite‐volume schemes for Friedrichs systems in multiple space dimensions: A priori and a posteriori error estimates

We consider a class of finite‐volume schemes on unstructured meshes for symmetric hyperbolic linear systems of balance laws in two and three space dimensions. This class of schemes has been introduced and analyzed by Vila and Villedieu ( 5 ). They have proven an a priori error estimate for approximations of smooth solutions. We extend the results to weak solutions. This is the base to derive an a posteriori error estimate for finite‐volume approximations of weak solutions. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A posteriori error estimate for finite volume element method of the parabolic equations

In this article, residual‐type a posteriori error estimates are studied for finite volume element (FVE) method of parabolic equations. Residual‐type a posteriori error estimator is constructed and the reliable and efficient bounds for the error estimator are established. Residual‐type a posteriori error estimator can be used to assess the accuracy of the FVE solutions in practical applications. Some numerical examples are provided to confirm the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 259–275, 2017     

Adaptive numerical solution of thick plates using first‐order shear deformation theory. Part II: Adaptivity

This is the second in a pair of articles concerned with the adaptive finite element solution of Riessner‐Mindlin thick plates modeled using first‐order shear deformation theory. This article is concerned with enhancing the a posteriori energy‐error estimators developed in Part I in order to accomodate transition elements in the finite element mesh. The resulting estimators are then used in an adaptive finite element model employing transition elements and the subsequent results discussed and compared with those in Part I. A major part of the article is devoted to identifying a novel patch assembly node algorithm for using the ZZ recovery‐type estimator with transition elements. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 227–253, 2003.     

A posteriori error estimates of edge residual type of finite element method for nonmonotone quasi‐linear elliptic problems

In this article, we study the edge residual‐based a posteriori error estimates of conforming linear finite element method for nonmonotone quasi‐linear elliptic problems. It is proven that edge residuals dominate a posteriori error estimates. Up to higher order perturbations, edge residuals can act as a posteriori error estimators. The global reliability and local efficiency bounds are established both in H ¹‐norm and L ²‐norm. Numerical experiments are provided to illustrate the performance of the proposed error estimators. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 813–837, 2014     

An optimal‐order error estimate for a Galerkin‐mixed finite‐element time‐stepping procedure for porous media flows

This article deals with the numerical approximation of miscible displacement problem of one incompressible fluid in a porous medium. The adopted formulation is based on the combined use of a mixed finite‐element scheme to treat pressure equation and of the finite‐element approach to treat concentration equation. Optimal‐order error estimates are obtained under some milder mesh‐parameter constraints. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 707–719, 2012     

Connections between discontinuous Galerkin and nonconforming finite element methods for the Stokes equations

We study a discontinuous Galerkin finite element method (DGFEM) for the Stokes equations with a weak stabilization of the viscous term. We prove that, as the stabilization parameter γ tends to infinity, the solution converges at speed γ^?1 to the solution of some stable and well‐known nonconforming finite element methods (NCFEM) for the Stokes equations. In addition, we show that an a posteriori error estimator for the DGFEM‐solution based on the reconstruction of a locally conservative H(div, Ω)‐tensor tends at the same speed to a classical a posteriori error estimator for the NCFEM‐solution. These results can be used to affirm the robustness of the DGFEM‐method and also underline the close relationship between the two approaches. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A decoupling two‐grid algorithm for the mixed Stokes‐Darcy model with the Beavers‐Joseph interface condition

In this article, a decoupling scheme based on two‐grid finite element for the mixed Stokes‐Darcy problem with the Beavers‐Joseph interface condition is proposed and investigated. With a restriction of a physical parameter α, we derive the numerical stability and error estimates for the scheme. Numerical experiments indicate that such two‐grid based decoupling finite element schemes are feasible and efficient. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1066–1082, 2014     

Projection stabilized nonconforming finite element methods for the stokes problem

In this article we study a projection‐stabilized nonconforming finite element discretization of the Stokes problem. We present a priori error analysis and give a recovery‐based a posteriori error estimator for the considered problem. Numerical results illustrate the theoretical performance of the error estimator. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 218–240, 2017