Parallel Galerkin domain decomposition procedures based on the streamline diffusion method for convection-diffusion problems

Based upon the streamline diffusion method, parallel Galerkin domain decomposition procedures for convection-diffusion problems are given. These procedures use implicit method in the sub-domains and simple explicit flux calculations on the inter-boundaries of sub-domains by integral mean method or extrapolation method to predict the inner-boundary conditions. Thus, the parallelism can be achieved by these procedures. The explicit nature of the flux calculations induces a time step limitation that is necessary to preserve stability. Artificial diffusion parameters δ are given. By analysis, optimal order error estimate is derived in a norm which is stronger than L²-norm for these procedures. This error estimate not only includes the optimal H¹-norm error estimate, but also includes the error estimate along the streamline direction ‖β⋅∇(u−U)‖, which cannot be achieved by standard finite element method. Experimental results are presented to confirm theoretical results.     

Parallel Galerkin domain decomposition procedures for wave equation

Parallel Galerkin domain decomposition procedures for wave equation are given. These procedures use implicit method in the sub-domains and simple explicit flux calculation on the inter-boundaries of sub-domains by integral mean method or extrapolation method. Thus, the parallelism can be achieved by these procedures. The explicit nature of the flux prediction induces a time step constraint that is necessary to preserve the stability. L²-norm error estimates are derived for these procedures. Experimental results are presented to confirm the theoretical results.     

A Schwarz domain decomposition method with gradient projection for optimal control governed by elliptic partial differential equations

A domain decomposition method (DDM) is presented to solve the distributed optimal control problem. The optimal control problem essentially couples an elliptic partial differential equation with respect to the state variable and a variational inequality with respect to the constrained control variable. The proposed algorithm, called SA-GP algorithm, consists of two iterative stages. In the inner loops, the Schwarz alternating method (SA) is applied to solve the state and co-state variables, and in the outer loops the gradient projection algorithm (GP) is adopted to obtain the control variable. Convergence of iterations depends on both the outer and the inner loops, which are coupled and affected by each other. In the classical iteration algorithms, a given tolerance would be reached after sufficiently many iteration steps, but more iterations lead to huge computational cost. For solving constrained optimal control problems, most of the computational cost is used to solve PDEs. In this paper, a proposed iterative number independent of the tolerance is used in the inner loops so as to save a lot of computational cost. The convergence rate of L²-error of control variable is derived. Also the analysis on how to choose the proposed iteration number in the inner loops is given. Some numerical experiments are performed to verify the theoretical results.     

A posteriori error estimates for mixed finite element approximations of elliptic problems

We derive residual based a posteriori error estimates of the flux in L ²-norm for a general class of mixed methods for elliptic problems. The estimate is applicable to standard mixed methods such as the Raviart–Thomas–Nedelec and Brezzi–Douglas–Marini elements, as well as stabilized methods such as the Galerkin-Least squares method. The element residual in the estimate employs an elementwise computable postprocessed approximation of the displacement which gives optimal order.     

A posteriori error estimates for elliptic problems with Dirac delta source terms

The aim of this paper is to introduce residual type a posteriori error estimators for a Poisson problem with a Dirac delta source term, in L ^p norm and W^1,p seminorm. The estimators are proved to yield global upper and local lower bounds for the corresponding norms of the error. They are used to guide adaptive procedures, which are experimentally shown to lead to optimal orders of convergence.     

Finite element methods with numerical quadrature for elliptic problems with smooth interfaces

The purpose of this paper is to study the effect of the numerical quadrature on the finite element approximation to the exact solution of elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998) 175-202]. We derive error estimates in finite element method with quadrature for elliptic interface problems in a two-dimensional convex polygonal domain. Optimal order error estimates in L² and H¹ norms are shown to hold even if the regularity of the solution is low on the whole domain. Finally, numerical experiment for two dimensional test problem is presented in support of our theoretical findings.     

Optimal error estimation for Petrov-Galerkin methods in two dimensions

Summary We examine the optimality of conforming Petrov-Galerkin approximations for the linear convection-diffusion equation in two dimensions. Our analysis is based on the Riesz representation theorem and it provides an optimal error estimate involving the smallest possible constantC. It also identifies an optimal test space, for any choice of consistent norm, as that whose image under the Riesz representation operator is the trial space. By using the Helmholtz decomposition of the Hilbert space [L ²()]², we produce a construction for the constantC in which the Riesz representation operator is not required explicitly. We apply the technique to the analysis of the Galerkin approximation and of an upwind finite element method.     

DOMAIN DECOMPOSITION METHOD FOR PARABOLIC EQUATION ON GENERAL DOMAIN

The nonoxerlapping domain deoomposition method for parabolic partial differential equation on general domain is considered. A kind of domain decomposition that uses the finite element procedure ks given. The problem.over the domains can be implemented on parallel computer. Convergence analysis is also presented.     

Optimal error estimate and superconvergence of the DG method for first-order hyperbolic problems

We consider the original discontinuous Galerkin method for the first-order hyperbolic problems in d-dimensional space. We show that, when the method uses polynomials of degree k, the L₂-error estimate is of order k+1 provided the triangulation is made of rectangular elements satisfying certain conditions. Further, we show the O(h^2k+1)-order superconvergence for the error on average on some suitably chosen subdomains (including the whole domain) and their outflow faces. Moreover, we also establish a derivative recovery formula for the approximation of the convection directional derivative which is superconvergent with order k+1.     

Domain decomposition and splitting methods for Mortar mixed finite element approximations to parabolic equations

Summary We introduce in this article a new domain decomposition algorithm for parabolic problems that combines Mortar Mixed Finite Element methods for the space discretization with operator splitting schemes for the time discretization. The main advantage of this method is to be fully parallel. The algorithm is proven to be unconditionally stable and a convergence result in (Δt/h ^1/2) is presented.     

Parallel defect control

How can small-scale parallelism best be exploited in the solution of nonstiff initial value problems? It is generally accepted that only modest gains inefficiency are possible, and it is often the case that “fast” parallel algorithms have quite crude error control and stepsize selection components. In this paper we consider the possibility of using parallelism to improvereliability andfunctionality rather than efficiency. We present an algorithm that can be used with any explicit Runge-Kutta formula. The basic idea is to take several smaller substeps in parallel with the main step. The substeps provide an interpolation facility that is essentially free, and the error control strategy can then be based on a defect (residual) sample. If the number of processors exceeds (p ? 1)/2, wherep is the order of the Runge-Kutta formula, then the interpolant and the error control scheme satisfy very strong reliability conditions. Further, for a given orderp, the asymptotically optimal values for the substep lengths are independent of the problem and formula and hence can be computed a priori. Theoretical comparisons between the parallel algorithm and optimal sequential algorithms at various orders are given. We also report on numerical tests of the reliability and efficiency of the new algorithm, and give some parallel timing statistics from a 4-processor machine.     

A posteriori error estimates for hp finite element solutions of convex optimal control problems

In this paper, we present a posteriori error analysis for hp finite element approximation of convex optimal control problems. We derive a new quasi-interpolation operator of Clément type and a new quasi-interpolation operator of Scott-Zhang type that preserves homogeneous boundary condition. The Scott-Zhang type quasi-interpolation is suitable for an application in bounding the errors in L²-norm. Then hp a posteriori error estimators are obtained for the coupled state and control approximations. Such estimators can be used to construct reliable adaptive finite elements for the control problems.     

The finite element method with non-uniform mesh sizes applied to the exterior Helmholtz problem

Summary The finite element method with non-uniform mesh sizes is employed to approximately solve Helmholtz type equations in unbounded domains. The given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary. This approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases. This results in a great reduction in the number of equations to be solved. It is proved that optimal error estimates hold inL ²,H ¹ andL , provided that certain relationships hold between the frequency, mesh size and outer radius.     

Convergence estimates for an higher order optimized Schwarz method for domains with an arbitrary interface

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. Optimized Schwarz methods employ a first or higher order boundary condition along the artificial interface to accelerate convergence. In the literature, the analysis of optimized Schwarz methods relies on Fourier analysis and so the domains are restricted to be regular (rectangular). In this paper, we express the interface operator of an optimized Schwarz method in terms of Poincare-Steklov operators. This enables us to derive an upper bound of the spectral radius of the operator arising in this method of 1−O(h^1/4) on a class of general domains, where h is the discretization parameter. This is the predicted rate for a second order optimized Schwarz method in the literature on rectangular subdomains and is also the observed rate in numerical simulations.     

Nonconforming mixed finite element approximation to the stationary Navier-Stokes equations on anisotropic meshes

The main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary Navier-Stokes equations under anisotropic meshes. That is, the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure. The convergence analysis is presented and the optimal error estimates both in a broken H¹-norm for the velocity and in an L²-norm for the pressure are derived on anisotropic meshes.     

A NONCONFORMING QUADRILATERAL FINITE ELEMENT APPROXIMATION TO NONLINEAR SCHRDINGER EQUATION

In this article, a nonconforming quadrilateral element(named modified quasiWilson element) is applied to solve the nonlinear schr¨odinger equation(NLSE). On the basis of a special character of this element, that is, its consistency error is of order O(h~3) for broken H1-norm on arbitrary quadrilateral meshes, which is two order higher than its interpolation error, the optimal order error estimate and superclose property are obtained. Moreover,the global superconvergence result is deduced with the help of interpolation postprocessing technique. Finally, some numerical results are provided to verify the theoretical analysis.     

Nonoverlapping domain decomposition characteristic finite differences for three‐dimensional convection‐diffusion equations

One domain decomposition method modified with characteristic differences is presented for non‐periodic three‐dimensional equations by multiply‐type quadratic interpolation and variant time‐step technique. This method consists of reduced‐scale, two‐dimensional computation on subdomain interface boundaries and fully implicit subdomain computation in parallel. A computational algorithm is outlined and an error estimate in discrete l²‐ norm is established by introducing new inner products and norms. Finally, numerical examples are given to illustrate the theoretical results, efficiency and parallelism of this method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 17‐37, 2012     

Biquadratic finite volume element methods based on optimal stress points for parabolic problems

In this paper, the semi-discrete and full discrete biquadratic finite volume element schemes based on optimal stress points for a class of parabolic problems are presented. Optimal order error estimates in H¹ and L² norms are derived. In addition, the superconvergences of numerical gradients at optimal stress points are also discussed. A numerical experiment confirms some results of theoretical analysis.     

Optimal recovery of 3D X-ray tomographic data via shearlet decomposition

This paper introduces a new decomposition of the 3D X-ray transform based on the shearlet representation, a multiscale directional representation which is optimally efficient in handling 3D data containing edge singularities. Using this decomposition, we derive a highly effective reconstruction algorithm yielding a near-optimal rate of convergence in estimating piecewise smooth objects from 3D X-ray tomographic data which are corrupted by white Gaussian noise. This algorithm is achieved by applying a thresholding scheme on the 3D shearlet transform coefficients of the noisy data which, for a given noise level ε, can be tuned so that the estimator attains the essentially optimal mean square error rate O(log(ε ^???1)ε ^2/3), as ε→0. This is the first published result to achieve this type of error estimate, outperforming methods based on Wavelet-Vaguelettes decomposition and on SVD, which can only achieve MSE rates of O(ε ^1/2) and O(ε ^1/3), respectively.     

Adaptive finite elements for exterior domain problems

Summary. We present an adaptive finite element method for solving elliptic problems in exterior domains, that is for problems in the exterior of a bounded closed domain in , . We describe a procedure to generate a sequence of bounded computational domains , , more precisely, a sequence of successively finer and larger grids, until the desired accuracy of the solution is reached. To this end we prove an a posteriori error estimate for the error on the unbounded domain in the energy norm by means of a residual based error estimator. Furthermore we prove convergence of the adaptive algorithm. Numerical examples show the optimal order of convergence. Received July 8, 1997 /Revised version received October 23, 1997