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H1 Norm (h1 + norm)

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Selected Abstracts

Some observations on the l2 convergence of the additive Schwarz preconditioned GMRES method

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2002
Xiao-Chuan Cai
Abstract Additive Schwarz preconditioned GMRES is a powerful method for solving large sparse linear systems of equations on parallel computers. The algorithm is often implemented in the Euclidean norm, or the discrete l2 norm, however, the optimal convergence result is available only in the energy norm (or the equivalent Sobolev H1 norm). Very little progress has been made in the theoretical understanding of the l2 behaviour of this very successful algorithm. To add to the difficulty in developing a full l2 theory, in this note, we construct explicit examples and show that the optimal convergence of additive Schwarz preconditioned GMRES in l2 cannot be obtained using the existing GMRES theory. More precisely speaking, we show that the symmetric part of the preconditioned matrix, which plays a role in the Eisenstat,Elman,Schultz theory, has at least one negative eigenvalue, and we show that the condition number of the best possible eigenmatrix that diagonalizes the preconditioned matrix, key to the Saad,Schultz theory, is bounded from both above and below by constants multiplied by h,1/2. Here h is the finite element mesh size. The results presented in this paper are mostly negative, but we believe that the techniques used in our proofs may have wide applications in the further development of the l2 convergence theory and in other areas of domain decomposition methods. Copyright © 2002 John Wiley & Sons, Ltd. [source]

An upwind finite volume element method based on quadrilateral meshes for nonlinear convection-diffusion problems

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2009
Fu-Zheng Gao
Abstract Considering an upwind finite volume element method based on convex quadrilateral meshes for computing nonlinear convection-diffusion problems, some techniques, such as calculus of variations, commutating operator, and the theory of prior error estimates and techniques, are adopted. Discrete maximum principle and optimal-order error estimates in H1 norm for fully discrete method are derived to determine the errors in the approximate solution. Thus, the well-known problem [(Li et al., Generalized difference methods for differential equations: numerical analysis of finite volume methods, Marcel Dekker, New York, 2000), p 365.] has been solved. Some numerical experiments show that the method is a very effective engineering computing method for avoiding numerical dispersion and nonphysical oscillations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009 [source]

Error analysis of the L2 least-squares finite element method for incompressible inviscid rotational flows,

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2004
Chiung-Chiou Tsai
Abstract In this article we analyze the L2 least-squares finite element approximations to the incompressible inviscid rotational flow problem, which is recast into the velocity-vorticity-pressure formulation. The least-squares functional is defined in terms of the sum of the squared L2 norms of the residual equations over a suitable product function space. We first derive a coercivity type a priori estimate for the first-order system problem that will play the crucial role in the error analysis. We then show that the method exhibits an optimal rate of convergence in the H1 norm for velocity and pressure and a suboptimal rate of convergence in the L2 norm for vorticity. A numerical example in two dimensions is presented, which confirms the theoretical error estimates. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004 [source]

An economical difference scheme for heat transport equation at the microscale,

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2004
Zhiyue Zhang
Abstract Heat transport at the microscale is of vital importance in microtechnology applications. In this article, we proposed a new ADI difference scheme of the Crank-Nicholson type for heat transport equation at the microscale. It is shown that the scheme is second order accurate in time and in space in the H1 norm. Numerical result implies that the theoretical analysis is correct and the scheme is effective. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004 [source]

Lagrange interpolation and finite element superconvergence,

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2004
Bo Li
Abstract We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d -dimensional Qk -type elements with d , 1 and k , 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H1 norm. For d -dimensional Pk -type elements, we consider the standard Lagrange interpolation,the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d , 2 and k , d + 1 that such interpolation and the finite element solution are not superclose in both H1 and L2 norms and that not all such interpolation points are superconvergence points for the finite element approximation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 33,59, 2004. [source]