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 domain decomposition procedures of improved D-D type for parabolic problems

Two parallel domain decomposition procedures for solving initial-boundary value problems of parabolic partial differential equations are proposed. One is the extended D-D type algorithm, which extends the explicit/implicit conservative Galerkin domain decomposition procedures, given in [5], from a rectangle domain and its decomposition that consisted of a stripe of sub-rectangles into a general domain and its general decomposition with a net-like structure. An almost optimal error estimate, without the factor H^−1/2 given in Dawson-Dupont’s error estimate, is proved. Another is the parallel domain decomposition algorithm of improved D-D type, in which an additional term is introduced to produce an approximation of an optimal error accuracy in L²-norm.     

Parallel Galerkin domain decomposition procedures for parabolic equation on general domain

Parallel Galerkin domain decomposition procedures for parabolic equation on general domain are given. These procedures use implicit Galerkin method in the subdomains and simple explicit flux calculation on the interdomain boundaries by integral mean method or extrapolation method to predict the inner‐boundary conditions. Thus, the parallelism can be achieved by these procedures. These procedures are conservative both in the subdomains and across interboundaries. The explicit nature of the flux prediction induces a time‐step limitation that is necessary to preserve stability, but this constraint is less severe than that for a fully explicit method. L²‐norm error estimates are derived for these procedures. Compared with the work of Dawson and Dupont [Math Comp 58 (1992), 21–35], these L²‐norm error estimates avoid the loss of H^?1/2 factor. Experimental results are presented to confirm the theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Fourier analysis of schwarz alternating methods for piecewise hermite bicubic orthogonal spline collocation

The rates of convergence of two Schwarz alternating methods are analyzed for the iterative solution of a discrete problem which arises when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet problem for Poisson's equation on a rectangle. In the first method, the rectangle is divided into two overlapping subrectangles, while three overlapping subrectangles are used in the second method. Fourier analysis is used to obtain explicit formulas for the convergence factors by which theH ¹-norm of the errors is reduced in one iteration of the Schwarz methods. It is shown numerically that while these factors depend on the size of overlap, they are independent of the partition stepsize. Results of numerical experiments are presented which confirm the established rates of convergence of the Schwarz methods.This research was supported in part by funds from the National Science Foundation grant CCR-9103451.     

Error estimates of the lumped mass finite element method for semilinear elliptic problems

We are concerned with the semilinear elliptic problems. We first investigate the L²-error estimate for the lumped mass finite element method. We then use the cascadic multigrid method to solve the corresponding discrete problem. On the basis of the finite element error estimates, we prove the optimality of the proposed multigrid method. We also report some numerical results to support the theory.     

A posteriori error estimation of residual type for anisotropic diffusion-convection-reaction problems

This paper presents a robust a posteriori residual error estimator for diffusion-convection-reaction problems with anisotropic diffusion, approximated by a SUPG finite element method on isotropic or anisotropic meshes in R^d, d=2 or 3. The equivalence between the energy norm of the error and the residual error estimator is proved. Numerical tests confirm the theoretical results.     

Superconvergence analysis of approximate boundary-flux calculations

Summary Certain projection post-processing techniques have been proposed for computing the boundary flux for two-dimensional problems (e.g., see Carey, et al. [5]). In a series of numerical experiments on elliptic problems they observed that these post-processing formulas for approximate fluxes were almost (O(h ²)-accurate for linear triangular elements. In this paper we prove that the computed boundary flux isO(h ² ln 1/h)-accurate in the maximum norm for the partial method of [5]. If the solutionuH ³() then the boundary flux error isO(h ^3/2) in theL ²-norm.     

A two-grid method for finite volume element approximations of second-order nonlinear hyperbolic equations

The two-grid method is studied for solving a two-dimensional second-order nonlinear hyperbolic equation using finite volume element method. The method is based on two different finite element spaces defined on one coarse grid with grid size H and one fine grid with grid size h, respectively. The nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step. It is proved that the coarse grid can be much coarser than the fine grid. A prior error estimate in the H¹-norm is proved to be O(h+H³|lnH|) for the two-grid semidiscrete finite volume element method. With these proposed techniques, solving such a large class of second-order nonlinear hyperbolic equations will not be much more difficult than solving one single linearized equation. Finally, a numerical example is presented to validate the usefulness and efficiency of the method.     

A posteriori error estimator based on gradient recovery by averaging for discontinuous Galerkin methods

We consider some (anisotropic and piecewise constant) diffusion problems in domains of R², approximated by a discontinuous Galerkin method with polynomials of any fixed degree. We propose an a posteriori error estimator based on gradient recovery by averaging. It is shown that this estimator gives rise to an upper bound where the constant is one up to some additional terms that guarantee reliability. The lower bound is also established. Moreover these additional terms are negligible when the recovered gradient is superconvergent. The reliability and efficiency of the proposed estimator is confirmed by some numerical tests.     

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.     

Local flux mimetic finite difference methods

We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes. To approximate the velocity (vector variable), the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n-gonal mesh face in three dimensions. To approximate the pressure (scalar variable), the method uses one degree of freedom per element. A specially chosen quadrature rule for the L ²-product of vector-functions allows for a local flux elimination and reduction of the method to a cell-centered finite difference scheme for the pressure unknowns. Under certain assumptions, first-order convergence is proved for both variables and second-order convergence is proved for the pressure. The assumptions are verified on simplicial meshes for a particular quadrature rule that leads to a symmetric method. For general polyhedral meshes, non-symmetric methods are constructed based on quadrature rules that are shown to satisfy some of the assumptions. Numerical results confirm the theory.     

A locking-free nonconforming triangular element for planar elasticity with pure traction boundary condition

A new nonconforming triangular element for the equations of planar linear elasticity with pure traction boundary conditions is considered. By virtue of construction of the element, the discrete version of Korn’s second inequality is directly proved to be valid. Convergence rate of the finite element methods is uniformly optimal with respect to λ. Error estimates in the energy norm and L²-norm are O(h²) and O(h³), respectively.     

Galerkin and subspace decomposition methods in space and time for the Navier-Stokes equations

The Galerkin method and the subspace decomposition method in space and time for the two-dimensional incompressible Navier-Stokes equations with the H²-initial data are considered. The subspace decomposition method consists of splitting the approximate solution as the sum of a low frequency component discretized by the small time step Δt and a high frequency one discretized by the large time step pΔt with p>1. The H²-stability and L²-error analysis for the subspace decomposition method are obtained. Finally, some numerical tests to confirm the theoretical results are provided.     

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.     

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.     

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.     

An approximation of incompressible miscible displacement in porous media by mixed finite element method and characteristics-mixed finite element method

An approximation scheme is defined for incompressible miscible displacement in porous media. This scheme is constructed by using two methods. Standard mixed finite element is used for the Darcy velocity equation. A characteristics-mixed finite element method is presented for the concentration equation. Characteristic approximation is applied to handle the convection part of the concentration equation, and a lowest-order mixed finite element spatial approximation is adopted to deal with the diffusion part. Thus, the scalar unknown concentration and the diffusive flux can be approximated simultaneously. In order to derive the optimal L²$L^2$-norm error estimates, a post-processing step is included in the approximation to the scalar unknown concentration. This scheme conserves mass globally; in fact, on the discrete level, fluid is transported along the approximate characteristics. Numerical experiments are presented finally to validate the theoretical analysis.     

A new high accuracy locally one-dimensional scheme for the wave equation

In this paper, a new locally one-dimensional (LOD) scheme with error of O(Δt⁴+h⁴) for the two-dimensional wave equation is presented. The new scheme is four layer in time and three layer in space. One main advantage of the new method is that only tridiagonal systems of linear algebraic equations have to be solved at each time step. The stability and dispersion analysis of the new scheme are given. The computations of the initial and boundary conditions for the two intermediate time layers are explicitly constructed, which makes the scheme suitable for performing practical simulation in wave propagation modeling. Furthermore, a comparison of our new scheme and the traditional finite difference scheme is given, which shows the superiority of our new method.     

A penalty finite element method based on the Euler implicit/explicit scheme for the time-dependent Navier-Stokes equations

A fully discrete penalty finite element method is presented for the two-dimensional time-dependent Navier-Stokes equations, where the time discretization is based on the Euler implicit/explicit scheme with some implicit linear terms and an explicit nonlinear term, and the finite element spatial discretization is based on the P₁b-P₁ element pair, which satisfies the discrete inf-sup condition. This method allows us to separate the computation of the velocity from the computation of the pressure with a larger time-step size Δt, so that the numerical velocity and the pressure are easily computed. An optimal error estimate of the numerical velocity and the pressure is provided for the fully discrete penalty finite element method when the penalty parameter ?, the time-step size Δt and the mesh size h satisfy the following stability conditions: ?c₁≤1, Δtκ₁≤1 and h²≤β₁Δt, respectively, for some positive constants c₁, κ₁ and β₁. Finally, some numerical tests to confirm the theoretical results of the penalty finite element method are provided.