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Many numerical methods used to solve Ordinary Differential Equations, or Differential Algebraic Equations can be written as general linear methods. The B-convergence results for general linear methods are for algebraically stable methods, and ...

We investigate some classes of general linear methods withs internal andr external approximations, with stage orderq and orderp, adjacent to the class withs=r=q=p considered by Butcher. We demonstrate that interesting methods exist also ifs+1=r=q, ...

This paper deals with convergence results for a special class of Runge-Kutta (RK) methods as applied to differential-algebraic equations (DAE's) of index 2 in Hessenberg form. The considered methods are stiffly accurate, with a singular RK matrix ...

We develop an algorithm to find k , the minimal value of f ⁽²⁾ , where f C ¹ is a quadratic spline with free knots, which interpolates the given p points { x _i , y _i } ₁ ^p with increasing x _i 's, has absolutely continuous f ⁽¹⁾ ...

The possibility of balancing the iteration and discretization errors in iterative solution of large systems of initial value problems is discussed. The main result answers the question affirmatively by stating that in the convergence process, in a model ...

In order to solve the Stokes equations numerically, Crouzeix and Raviart introduced elements satisfying a discrete divergence condition. For the two dimensional case and uniform triangulations it is shown, that using the standard basis functions, the ...

A uniform framework for the study of upwinding schemes is developed. The standard finite element Galerkin discretization is chosen as the reference discretization, and differences between other discretization schemes and the reference are written as ...

The hierarchical basis preconditioner and the recent preconditioner of Bramble, Pasciak and Xu are derived and analyzed within a joint framework. This discussion elucidates the close relationship between both methods. Special care is devoted to highly ...

We study symmetric positive definite linear systems, with a 2-by-2 block matrix preconditioned by inverting directly one of the diagonal blocks and suitably preconditioning the other. Using an approximate version of Young's "Property A", we show that ...

The study of the finite element approximation to nonlinear second order elliptic boundary value problems with discontinuous coefficients is presented in the case of mixed Dirichlet-Neumann boundary conditions. The change in domain and numerical ...

This paper is concerned with the (preconditioned) conjugate gradient method for solving systems of linear algebraic equationsAx=b withA having Red/Black form. The connections between the Conjugate Gradient (CG) scheme applied toAx=b and its two ...

The concept of suitability means that the nonlinear equations to be solved in an implicit Runga-Kutta method have a unique solution. In this paper, we introduce the concept of D-suitability and show that previous results become special cases of ...

We present two a posteriori error estimators for the mini-element discretization of the Stokes equations. One is based on a suitable evaluation of the residual of the finite element solution. The other one is based on the solution of suitable local ...

The homotopy method can be used to solve eigenvalue-eigenvector problems. The purpose of this paper is to report the numerical experience of the homotopy method of computing eigenpairs for real symmetric tridiagonal matrices together with a couple of ...

The theory of Hermite-spline interpolation on the equidistant lattice Z is written in purly real terms and this for an arbitrary polynomial degree, Hermitian order and node-shift parameter. An explicit representation formula for the Hermitian ...

We report a modification of the Stiefel-Bettis method which is of trigonometric order one and of polynomial order two for the general second order initial value problems. We also discuss the modified Stiefel-Bettis method made explicit for the ...

AnO(h ⁶) collocation method based on quintic splines is developed and analyzed for general fourth-order linear two-point boundary value problems. The method determines a quintic spline approximation to the solution by forcing it to satisfy a high ...

A method of finding the closest normal matrix in the Frobenius matrix norm is developed. It is shown that if a matrix is represented in those coordinates where its closest normal matrix is diagonal, its restriction to any pair of coordinate ...

The numerical solution of Volterra integral equations of the first kind can be achieved via product integration. This paper establishes the asymptotic error expansions of certain product integration rules. The rectangular rules are found to ...

Second order methods for simultaneous approximation of multiple complex zeros of a polynomial are presented. Convergence analysis of new iteration formulas and an efficient criterion for the choice of the appropriate value of a root are discussed. ...