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## Error Estimates (error + estimate)
Kinds of Error Estimates
## Selected Abstracts## Error estimate and regularity for the compressible Navier-Stokes equations by Newton's method NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2003Sang Dong KimAbstract The finite element discretization error estimate and H1 regularity are shown for the solution generated by Newton's method to the stationary compressible Navier-Stokes equations by interpreting Newton's method as an equivalent iterative method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 511,524, 2003 [source] ## Error estimates in 2-node shear-flexible beam elements INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2003Gajbir SinghAbstract The objective of the paper is to report the investigation of error estimates/or convergence characteristics of shear-flexible beam elements. The order and magnitude of principal discretization error in the usage of various types beam elements such as: (a) 2-node standard isoparametric element, (b) 2-node field-consistent/reduced integration element and (c) 2-node coupled-displacement field element, is assessed herein. The method employs classical order of error analyses that is commonly used to evaluate the discretization error of finite difference methods. The finite element equilibrium equations at any node are expressed in terms of differential equations through the use of Taylor series. These differential equations are compared with the governing equations and error terms are identified. It is shown that the discretization error in coupled-field elements is the least compared to the field-consistent and standard finite elements (based on exact integration). Copyright © 2003 John Wiley & Sons, Ltd. [source] ## New stabilized finite element method for time-dependent incompressible flow problems INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2010*Article first published online: 20 FEB 200, Yueqiang ShangAbstract A new stabilized finite element method is considered for the time-dependent Stokes problem, based on the lowest-order P1,P0 and Q1,P0 elements that do not satisfy the discrete inf,sup condition. The new stabilized method is characterized by the features that it does not require approximation of the pressure derivatives, specification of mesh-dependent parameters and edge-based data structures, always leads to symmetric linear systems and hence can be applied to existing codes with a little additional effort. The stability of the method is derived under some regularity assumptions. Error estimates for the approximate velocity and pressure are obtained by applying the technique of the Galerkin finite element method. Some numerical results are also given, which show that the new stabilized method is highly efficient for the time-dependent Stokes problem. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Error estimates of CFVE method for fully nonlinear convection-dominated diffusion problems MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2010Yang ZhangAbstract Finite volume method and characteristics finite element method are two important methods for solving the partial differential equations. These two methods are combined in this paper to establish a fully discrete characteristics finite volume method for fully nonlinear convection-dominated diffusion problems. Through detailed theoretical analysis, optimal order H1 norm error estimates are obtained for this fully discrete scheme. Copyright © 2010 John Wiley & Sons, Ltd. [source] ## Error estimates for mixed finite element approximations of the viscoelasticity wave equation MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 17 2004Liping GaoAbstract This paper studies mixed finite element approximations to the solution of the viscoelasticity wave equation. Two new transformations are introduced and a corresponding system of first-order differential-integral equations is derived. The semi-discrete and full-discrete mixed finite element methods are then proposed for the problem based on the Raviart,Thomas,Nedelec spaces. The optimal error estimates in L2 -norm are obtained for the semi-discrete and full-discrete mixed approximations of the general viscoelasticity wave equation. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## Lp error estimates and superconvergence for covolume or finite volume element methods NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2003So-Hsiang ChouAbstract We consider convergence of the covolume or finite volume element solution to linear elliptic and parabolic problems. Error estimates and superconvergence results in the Lp norm, 2 , p , ,, are derived. We also show second-order convergence in the Lp norm between the covolume and the corresponding finite element solutions and between their gradients. The main tools used in this article are an extension of the "supercloseness" results in Chou and Li [Math Comp 69(229) (2000), 103,120] to the Lp based spaces, duality arguments, and the discrete Green's function method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 463,486, 2003 [source] ## Semilinear parabolic problem with nonstandard boundary conditions: Error estimates NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2003Marián SlodiAbstract We study a semilinear parabolic partial differential equation of second order in a bounded domain , , ,N, with nonstandard boundary conditions (BCs) on a part ,non of the boundary ,,. Here, neither the solution nor the flux are prescribed pointwise. Instead, the total flux through ,non is given, and the solution along ,non has to follow a prescribed shape function, apart from an additive (unknown) space-constant ,(t). We prove the well-posedness of the problem, provide a numerical method for the recovery of the unknown boundary data, and establish the error estimates. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 167,191, 2003 [source] ## hp -Adaptive Finite Element Methods for Variational Inequalities PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2008Andreas SchröderArticle first published online: 25 FEB 200In this work, we combine an hp,adaptive strategy with a posteriori error estimates for variational inequalities, which are given by contact problems. The a posteriori error estimates are obtained using a general approach based on the saddle point formulation of contact problems and making use of a yposteriori error estimates for variational equations. Error estimates are presented for obstacle problems and Signorini problems with friction. Numerical experiments confirm the reliability of the error estimates for finite elements of higher order. The use of the hp,adaptive strategy leads to meshes with the same characteristics as geometric meshes and to exponential convergence. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## On the convergence rate of vanishing viscosity approximations COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 8 2004Alberto BressanGiven a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ,u(t, ·) , u,(t, ·), = O(1)(1 + t) · |ln ,| on the distance between an exact BV solution u and a viscous approximation u,, letting the viscosity coefficient , , 0. In the proof, starting from u we construct an approximation of the viscous solution u, by taking a mollification u * and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed ,. Error estimates are then obtained by introducing new Lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families. © 2004 Wiley Periodicals, Inc. [source] ## On semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 7 2004Alexander TovbisWe calculate the leading-order term of the solution of the focusing nonlinear (cubic) Schrödinger equation (NLS) in the semiclassical limit for a certain one-parameter family of initial conditions. This family contains both solitons and pure radiation. In the pure radiation case, our result is valid for all times t , 0. We utilize the Riemann-Hilbert problem formulation of the inverse scattering problem to obtain the leading-order term of the solution. Error estimates are provided. © 2004 Wiley Periodicals, Inc. [source] ## An Adaptive Strategy for the Local Discontinuous Galerkin Method Applied to Porous Media Problems COMPUTER-AIDED CIVIL AND INFRASTRUCTURE ENGINEERING, Issue 4 2008Esov S. VelázquezDG methods may be viewed as high-order extensions of the classical finite volume method and, since no interelement continuity is imposed, they can be defined on very general meshes, including nonconforming meshes, making these methods suitable for h-adaptivity. The technique starts with an initial conformal spatial discretization of the domain and, in each step, the error of the solution is estimated. The mesh is locally modified according to the error estimate by performing two local operations: refinement and agglomeration. This procedure is repeated until the solution reaches a desired accuracy. The performance of this technique is examined through several numerical experiments and results are compared with globally refined meshes in examples with known analytic solutions. [source] ## A Three-step Method for Choosing the Number of Bootstrap Repetitions ECONOMETRICA, Issue 1 2000Donald W. K. AndrewsThis paper considers the problem of choosing the number of bootstrap repetitions B for bootstrap standard errors, confidence intervals, confidence regions, hypothesis tests, p -values, and bias correction. For each of these problems, the paper provides a three-step method for choosing B to achieve a desired level of accuracy. Accuracy is measured by the percentage deviation of the bootstrap standard error estimate, confidence interval length, test's critical value, test's p -value, or bias-corrected estimate based on B bootstrap simulations from the corresponding ideal bootstrap quantities for which B=,. The results apply quite generally to parametric, semiparametric, and nonparametric models with independent and dependent data. The results apply to the standard nonparametric iid bootstrap, moving block bootstraps for time series data, parametric and semiparametric bootstraps, and bootstraps for regression models based on bootstrapping residuals. Monte Carlo simulations show that the proposed methods work very well. [source] ## A robust a priori error estimate for the Fortin,Soulie finite element method INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2005David J. BlackerArticle first published online: 14 MAR 200Abstract It is well known that conforming finite element schemes exhibit Poisson locking in the incompressible limit as the Poisson ratio , tends to 1/2. A remedy for this is to use a non-conforming method (Math. Comput. 1992; 59:321-328) in which an a priori error bound is proved for the Crouzeix,Raviart scheme. In this paper we derive a new a priori estimate for the error in energy for the Fortin,Soulie finite element method using a method similar to that used in Brenner and Sung. We then illustrate the new error bound by presenting some numerical examples. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far-field boundary treatment, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 10 2010I. KalashnikovaAbstract A reduced order model (ROM) based on the proper orthogonal decomposition (POD)/Galerkin projection method is proposed as an alternative discretization of the linearized compressible Euler equations. It is shown that the numerical stability of the ROM is intimately tied to the choice of inner product used to define the Galerkin projection. For the linearized compressible Euler equations, a symmetry transformation motivates the construction of a weighted L2 inner product that guarantees certain stability bounds satisfied by the ROM. Sufficient conditions for well-posedness and stability of the present Galerkin projection method applied to a general linear hyperbolic initial boundary value problem (IBVP) are stated and proven. Well-posed and stable far-field and solid wall boundary conditions are formulated for the linearized compressible Euler ROM using these more general results. A convergence analysis employing a stable penalty-like formulation of the boundary conditions reveals that the ROM solution converges to the exact solution with refinement of both the numerical solution used to generate the ROM and of the POD basis. An a priori error estimate for the computed ROM solution is derived, and examined using a numerical test case. Published in 2010 by John Wiley & Sons, Ltd. [source] ## Upper and lower bounds in limit analysis: Adaptive meshing strategies and discontinuous loading INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2009J. J. MuñozAbstract Upper and lower bounds of the collapse load factor are here obtained as the optimum values of two discrete constrained optimization problems. The membership constraints for Von Mises and Mohr,Coulomb plasticity criteria are written as a set of quadratic constraints, which permits one to solve the optimization problem using specific algorithms for Second-Order Conic Program (SOCP). From the stress field at the lower bound and the velocities at the upper bound, we construct a novel error estimate based on elemental and edge contributions to the bound gap. These contributions are employed in an adaptive remeshing strategy that is able to reproduce fan-type mesh patterns around points with discontinuous surface loading. The solution of this type of problems is analysed in detail, and from this study some additional meshing strategies are also described. We particularise the resulting formulation and strategies to two-dimensional problems in plane strain and we demonstrate the effectiveness of the method with a set of numerical examples extracted from the literature. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## Smooth finite element methods: Convergence, accuracy and properties INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2008Hung Nguyen-XuanAbstract A stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented. The integration of the stiffness matrix is performed on the boundaries of the finite elements. A rigorous variational framework based on the Hu,Washizu assumed strain variational form is developed. We prove that solutions yielded by the proposed method are in a space bounded by the standard, finite element solution (infinite number of subcells) and a quasi-equilibrium finite element solution (a single subcell). We show elsewhere the equivalence of the one-subcell element with a quasi-equilibrium finite element, leading to a global a posteriori error estimate. We apply the method to compressible and incompressible linear elasticity problems. The method can always achieve higher accuracy and convergence rates than the standard finite element method, especially in the presence of incompressibility, singularities or distorted meshes, for a slightly smaller computational cost. It is shown numerically that the one-cell smoothed four-noded quadrilateral finite element has a convergence rate of 2.0 in the energy norm for problems with smooth solutions, which is remarkable. For problems with rough solutions, this element always converges faster than the standard finite element and is free of volumetric locking without any modification of integration scheme. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## On the a priori and a posteriori error analysis of a two-fold saddle-point approach for nonlinear incompressible elasticity INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2006Gabriel N. GaticaAbstract In this paper, we reconsider the a priori and a posteriori error analysis of a new mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions. The approach, being based only on the fact that the resulting variational formulation becomes a two-fold saddle-point operator equation, simplifies the analysis and improves the results provided recently in a previous work. Thus, a well-known generalization of the classical Babu,ka,Brezzi theory is applied to show the well-posedness of the continuous and discrete formulations, and to derive the corresponding a priori error estimate. In particular, enriched PEERS subspaces are required for the solvability and stability of the associated Galerkin scheme. In addition, we use the Ritz projection operator to obtain a new reliable and quasi-efficient a posteriori error estimate. Finally, several numerical results illustrating the good performance of the associated adaptive algorithm are presented. Copyright © 2006 John Wiley & Sons, Ltd. [source] ## An a posteriori error estimator for the p - and hp -versions of the finite element method INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2005J. E. TarancónAbstract An a posteriori error estimator is proposed in this paper for the p - and hp -versions of the finite element method in two-dimensional linear elastostatic problems. The local error estimator consists in an enhancement of an error indicator proposed by Bertóti and Szabó (Int. J. Numer. Meth. Engng. 1998; 42:561,587), which is based on the minimum complementary energy principle. In order to obtain the local error estimate, this error indicator is corrected by a factor which depends only on the polynomial degree of the element. The proposed error estimator shows a good effectivity index in meshes with uniform and non-uniform polynomial distributions, especially when the global error is estimated. Furthermore, the local error estimator is reliable enough to guide p - and hp -adaptive refinement strategies. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## An adaptive non-conforming finite-element method for Reissner,Mindlin plates INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2003Carsten CarstensenAbstract Adaptive algorithms are important tools for efficient finite-element mesh design. In this paper, an error controlled adaptive mesh-refining algorithm is proposed for a non-conforming low-order finite-element method for the Reissner,Mindlin plate model. The algorithm is controlled by a reliable and efficient residual-based a posteriori error estimate, which is robust with respect to the plate's thickness. Numerical evidence for this and the efficiency of the new algorithm is provided in the sense that non-optimal convergence rates are optimally improved in our numerical experiments. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## A least square extrapolation method for the a posteriori error estimate of the incompressible Navier Stokes problem INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1 2005M. GarbeyAbstract A posteriori error estimators are fundamental tools for providing confidence in the numerical computation of PDEs. To date, the main theories of a posteriori estimators have been developed largely in the finite element framework, for either linear elliptic operators or non-linear PDEs in the absence of disparate length scales. On the other hand, there is a strong interest in using grid refinement combined with Richardson extrapolation to produce CFD solutions with improved accuracy and, therefore, a posteriori error estimates. But in practice, the effective order of a numerical method often depends on space location and is not uniform, rendering the Richardson extrapolation method unreliable. We have recently introduced (Garbey, 13th International Conference on Domain Decomposition, Barcelona, 2002; 379,386; Garbey and Shyy, J. Comput. Phys. 2003; 186:1,23) a new method which estimates the order of convergence of a computation as the solution of a least square minimization problem on the residual. This method, called least square extrapolation, introduces a framework facilitating multi-level extrapolation, improves accuracy and provides a posteriori error estimate. This method can accommodate different grid arrangements. The goal of this paper is to investigate the power and limits of this method via incompressible Navier Stokes flow computations. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Composite adaptive and input observer-based approaches to the cylinder flow estimation in spark ignition automotive engines INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, Issue 2 2004A. StotskyAbstract The performance of air charge estimation algorithms in spark ignition automotive engines can be enhanced using advanced estimation techniques available in the controls literature. This paper illustrates two approaches of this kind that can improve the cylinder flow estimation for gasoline engines without external exhaust gas recirculation (EGR). The first approach is based on an input observer, while the second approach relies on an adaptive estimator. Assuming that the cylinder flow is nominally estimated via a speed-density calculation, and that the uncertainty is additive to the volumetric efficiency, the straightforward application of an input observer provides an easy to implement algorithm that corrects the nominal air flow estimate. The experimental results that we report in the paper point to a sufficiently good transient behaviour of the estimator. The signal quality may deteriorate, however, for extremely fast transients. This motivates the development of an adaptive estimator that relies mostly on the feedforward speed-density calculation during transients, while during engine operation close to steady-state conditions, it relies mostly on the adaptation. In our derivation of the adaptive estimator, the uncertainty is modelled as an unknown parameter multiplying the intake manifold temperature. We use the tracking error between the measured and modelled intake manifold pressure together with an appropriately defined prediction error estimate to develop an adaptation algorithm with improved identifiability and convergence rate. A robustness enhancement, via a ,-modification with the ,-factor depending on the prediction error estimate, ensures that in transients the parameter estimate converges to a pre-determined a priori value. In close to steady-state conditions, the ,-modification is rendered inactive and the evolution of the parameter estimate is determined by both tracking error and prediction error estimate. Further enhancements are made by incorporating a functional dependence of the a priori value on the engine operating conditions such as the intake manifold pressure. The coefficients of this function can be learned during engine operation from the values to which the parameter estimate converges in close to steady-state conditions. This feedforward learning functionality improves transient estimation accuracy and reduces the convergence time of the parameter estimate. Copyright © 2004 John Wiley & Sons, Ltd. [source] ## hp -Mortar boundary element method for two-body contact problems with friction MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 17 2008Alexey ChernovAbstract We construct a novel hp -mortar boundary element method for two-body frictional contact problems for nonmatched discretizations. The contact constraints are imposed in the weak sense on the discrete set of Gauss,Lobatto points using the hp -mortar projection operator. The problem is reformulated as a variational inequality with the Steklov,Poincaré operator over a convex cone of admissible solutions. We prove an a priori error estimate for the corresponding Galerkin solution in the energy norm. Due to the nonconformity of our approach, the Galerkin error is decomposed into the approximation error and the consistency error. Finally, we show that the Galerkin solution converges to the exact solution as ,,((h/p)1/4) in the energy norm for quasiuniform discretizations under mild regularity assumptions. We solve the Galerkin problem with a Dirichlet-to-Neumann algorithm. The original two-body formulation is rewritten as a one-body contact subproblem with friction and a one-body Neumann subproblem. Then the original two-body frictional contact problem is solved with a fixed point iteration. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## A time-marching finite element method for an electromagnetic scattering problem MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 12 2003Tri VanAbstract In this paper, Newmark time-stepping scheme and edge elements are used to numerically solve the time-dependent scattering problem in a three-dimensional polyhedral cavity. Finite element methods based on the variational formulation derived in Van and Wood (Adv. Comput. Math., to appear) are considered. Existence and uniqueness of the discrete problem is proved by using Babuska,Brezzi theory. Finite element error estimate and stability of the Newmark scheme are also established. Copyright © 2003 John Wiley & Sons, Ltd. [source] ## A FEM,DtN formulation for a non-linear exterior problem in incompressible elasticity MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2003Gabriel N. GaticaAbstract 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. [source] ## Error norm estimation and stopping criteria in preconditioned conjugate gradient iterations NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2001Owe AxelssonAbstract Some techniques suitable for the control of the solution error in the preconditioned conjugate gradient method are considered and compared. The estimation can be performed both in the course of the iterations and after their termination. The importance of such techniques follows from the non-existence of some reasonable a priori error estimate for very ill-conditioned linear systems when sufficient information about the right-hand side vector is lacking. Hence, some a posteriori estimates are required, which make it possible to verify the quality of the solution obtained for a prescribed right-hand side. The performance of the considered error control procedures is demonstrated using real-world large-scale linear systems arising in computational mechanics. Copyright © 2001 John Wiley & Sons, Ltd. [source] ## Adaptive finite element approximations on nonmatching grids for second-order elliptic problems NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2010Hongsen ChenAbstract In this article we consider a finite element approximation for a model elliptic problem of second order on non-matching grids. This method combines the continuous finite element method with interior penalty discontinuous Galerkin method. As a special case, we develop a finite element method that is continuous on the matching part of the grid and is discontinuous on the nonmatching part. A residual type a posteriori error estimate is derived. Results of numerical experiments are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 [source] ## A CFL-free explicit characteristic interior penalty scheme for linear advection-reaction equations NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2010Kaixin WangAbstract We develop a CFL-free, explicit characteristic interior penalty scheme (CHIPS) for one-dimensional first-order advection-reaction equations by combining a Eulerian-Lagrangian approach with a discontinuous Galerkin framework. The CHIPS method retains the numerical advantages of the discontinuous Galerkin methods as well as characteristic methods. An optimal-order error estimate in the L2 norm for the CHIPS method is derived and numerical experiments are presented to confirm the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010 [source] ## Finite element analysis of thermally coupled nonlinear Darcy flows NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2010Jiang ZhuAbstract We consider a coupled system describing nonlinear Darcy flows with temperature dependent viscosity and with viscous heating. We first establish existence, uniqueness, and regularity of the weak solution of the system of equations. Next, we decouple the coupled system by a fixed point algorithm and propose its finite element approximation. Finally, we present convergence analysis with an error estimate between continuous solution and its iterative finite element approximation.© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010 [source] ## Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2010Hong-Lin LiaoAbstract Alternating direction implicit (ADI) schemes are computationally efficient and widely utilized for numerical approximation of the multidimensional parabolic equations. By using the discrete energy method, it is shown that the ADI solution is unconditionally convergent with the convergence order of two in the maximum norm. Considering an asymptotic expansion of the difference solution, we obtain a fourth-order, in both time and space, approximation by one Richardson extrapolation. Extension of our technique to the higher-order compact ADI schemes also yields the maximum norm error estimate of the discrete solution. And by one extrapolation, we obtain a sixth order accurate approximation when the time step is proportional to the squares of the spatial size. An numerical example is presented to support our theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 [source] ## A uniform optimal-order estimate for an Eulerian-Lagrangian discontinuous Galerkin method for transient advection,diffusion equations NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2009Kaixin WangArticle first published online: 11 APR 200Abstract We prove an optimal-order error estimate in a weighted energy norm for the Eulerian-Lagrangian discontinuous Galerkin method for unsteady-state advection,diffusion equations with general inflow and outflow boundary conditions. It is well-known that these problems admit dynamic fronts with interior and boundary layers. The estimate holds uniformly with respect to the vanishing diffusion coefficient. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source] |