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Based on the singular value decomposition, we obtain both additive and multiplicative perturbation bounds for the orthogonal projection, which improve some existing results. Furthermore, the Q-norm bounds for additive and multiplicative perturbations of the orthogonal projection are also given.
In this paper we develop isoparametric C0 interior penalty methods for plate bending problems on smooth domains. The orders of convergence of these methods are shown to be optimal in the energy norm. We also consider the convergence of these methods in lower order Sobolev norms and discuss subparametric C0 interior penalty methods. Numerical results that illustrate the performance of these methods...
In this paper, we consider an indefinite block triangular preconditioner for symmetric saddle point problems. The new eigenvalue distribution of the preconditioned matrix is derived and some corresponding results in Simoncini (Appl. Numer. Math. 49:63–80, 2004) and Wu et al. (Computing 84:183–208, 2009) are improved. Finally, numerical experiments of a model Stokes problem are reported.
In a previous article (Glowinski, J. Math. Anal. Appl. 41, 67–96, 1973) the first author discussed several methods for the numerical solution of nonlinear equations of the integro-differential type with periodic boundary conditions. In this article we discuss an alternative methodology largely based on the Strang’s symmetrized operator-splitting scheme. Several numerical experiments suggest that the...
We generalize the a posteriori techniques for the linear heat equation in Verfürth (Calcolo 40(3):195–212, 2003) to the case of the nonlinear parabolic $$p$$ -Laplace problem thereby proving reliable and efficient a posteriori error estimates for a fully discrete implicite Euler Galerkin finite element scheme. The error is analyzed using the so-called quasi-norm and a related dual error expression...
A Nyström method is proposed for solving systems of Fredholm integral equations equivalent to special boundary value problems of order $$2s$$ . The stability and the convergence of the proposed procedure is proved. The GMRes method is applied to solve the involved systems of linear equations. Some numerical examples are provided in order to illustrate the accuracy of the method.
The problems of analytic continuation are, in general, severely ill-posed. In this paper we proposed a modified kernel method to solve this problem on a strip domain. The convergence estimates with an appropriate choice of the regularization parameter are obtained. Some numerical tests show that the proposed methods are effective.
The main purpose of this paper is to give the numerical integration scheme for a fast Petrov–Galerkin method for solving the generalized airfoil equation, considered in a recent paper (Cai, J. Complex. 25:420–436, 2009). This scheme leads to a fully discrete sparse linear system. We show that it requires a nearly linear computational cost to get this system, and the approximate solution of the resulting...
In this article a priori error estimates are derived for the finite element discretization of optimal distributed control problems governed by the biharmonic operator. The state equation is discretized in primal mixed form using continuous piecewise biquadratic finite elements, while piecewise constant approximations are used for the control. The error estimates derived for the state variable as well...
In this paper we investigate block SSOR multisplittings. When the coefficient matrix is a block H-matrix or a (generalized) block strictly diagonally dominant matrix, the convergence of the parallel block SSOR multisplitting method for solving nonsingular linear systems is proved. Two numerical examples are given to illustrate the theoretical results.
We introduce a type of modified truncations of approximate approximation with Gaussian kernels, which can be applied to approximate functions on compact intervals. Our results improve the related results of Chen and Cao (Appl Math Comput 217:725–734, 2010), Müller and Varnhorn (J Approx Theory 145:171–181, 2007). Also, we construct approximate approximation operators in multivariate cases and obtain...
A low-order mimetic finite difference method for Reissner–Mindlin plate problems is considered. Together with the source problem, the free vibration and the buckling problems are investigated. Details about the scheme implementation are provided, and the numerical results on several different types of meshes are reported.
A jump-diffusion model for the pricing of options leads to a partial integro-differential equation (PIDE). Discretizing the PIDE by certain method, we get a sequence of systems of linear equations, where the coefficient matrices are Toeplitz matrices. In this paper, we decompose the coefficient matrix as the sum of a tridiagonal matrix and a near low-rank matrix, and approximate the near low-rank...
We consider the Cauchy problem for first order differential-functional equations. We present finite difference schemes to approximate viscosity solutions of this problem. The functional dependence in the equation is of the Hale type. It contains, as a particular case, equation with a retarded and deviated argument, and differential-integral equation. Numerical examples to illustrate the theory are...
In this paper, a kinetic method to compute approximative solutions of the one-dimensional open channel flow equations with a varying cross-sectional area and a varying bottom profile is proposed. The scheme preserves the steady states at rest, keeps the water height non-negative and is able to handle dry channel beds. These three properties are essential indicators for the quality of a numerical scheme...
We give new conditions under which the Drazin inverse of a modified matrix $$A-CD^DB$$ can be expressed in terms of the Drazin inverse of $$A$$ and its generalized Schur complement $$Z=D-BA^DC$$ , generalizing some recent results in the literature.
In this paper, we conjecture a formula for the value of the Pythagoras number for real multivariate sum of squares polynomials as a function of the (total or coordinate) degree and the number of variables. The conjecture is based on the comparison between the number of parameters and the number of conditions for a corresponding low-rank representation. This is then numerically verified for a number...
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