A class of ABS algorithms for Diophantine linear systems

Summary. Systems of integer linear (Diophantine) equations arise from various applications. In this paper we present an approach, based upon the ABS methods, to solve a general system of linear Diophantine equations. This approach determines if the system has a solution, generalizing the classical fundamental theorem of the single linear Diophantine equation. If so, a solution is found along with an integer Abaffian (rank deficient) matrix such that the integer combinations of its rows span the integer null space of the cofficient matrix, implying that every integer solution is obtained by the sum of a single solution and an integer combination of the rows of the Abaffian. We show by a counterexample that, in general, it is not true that any set of linearly independent rows of the Abaffian forms an integer basis for the null space, contrary to a statement by Egervary. Finally we show how to compute the Hermite normal form for an integer matrix in the ABS framework. Received July 9, 1999 / Revised version received May 8, 2000 / Published online May 4, 2001     

Spectral methods for initial boundary value problems with nonsmooth data

Summary. In this paper we consider hyperbolic initial boundary value problems with nonsmooth data. We show that if we extend the time domain to minus infinity, replace the initial condition by a growth condition at minus infinity and then solve the problem using a filtered version of the data by the Galerkin-Collocation method using Laguerre polynomials in time and Legendre polynomials in space, then we can recover pointwise values with spectral accuracy, provided that the actual solution is piecewise smooth. For this we have to perform a local smoothing of the computed solution. Received August 1, 1995 / Revised version received August 19, 1997     

Symmetric collocation methods for linear differential-algebraic boundary value problems

Summary. We present symmetric collocation methods for linear differential-algebraic boundary value problems without restrictions on the index or the structure of the differential-algebraic equation. In particular, we do not require a separation into differential and algebraic solution components. Instead, we use the splitting into differential and algebraic equations (which arises naturally by index reduction techniques) and apply Gau?-type (for the differential part) and Lobatto-type (for the algebraic part) collocation schemes to obtain a symmetric method which guarantees consistent approximations at the mesh points. Under standard assumptions, we show solvability and stability of the discrete problem and determine its order of convergence. Moreover, we show superconvergence when using the combination of Gau? and Lobatto schemes and discuss the application of interpolation to reduce the number of function evaluations. Finally, we present some numerical comparisons to show the reliability and efficiency of the new methods. Received September 22, 2000 / Revised version received February 7, 2001 / Published online August 17, 2001     

Collocation methods for solving linear differential-algebraic boundary value problems

Summary. We consider boundary value problems for linear differential-algebraic equations with variable coefficients with no restriction on the index. A well-known regularisation procedure yields an equivalent index one problem with d differential and a=n-d algebraic equations. Collocation methods based on the regularised BVP approximate the solution x by a continuous piecewise polynomial of degree k and deliver, in particular, consistent approximations at mesh points by using the Radau schemes. Under weak assumptions, the collocation problems are uniquely and stably solvable and, if the unique solution x is sufficiently smooth, convergence of order min {k+1,2k-1} and superconvergence at mesh points of order 2k-1 is shown. Finally, some numerical experiments illustrating these results are presented. Received October 1, 1999 / Revised version received April 25, 2000 / Published online December 19, 2000     

Mixed finite element for the linear plate problem: the Hermann-Miyoshi model revisited

Summary. In this paper we study the relationship between the Hermann-Miyoshi and the Ciarlet-Raviart formulations of the first biharmonic problem. This study will be based on a decomposition principle which will leads us to a new convergence analysis explaining some discrepancies between numerical results obtained with the first formulation on certain meshes and some theoretical convergence results. Received May 24, 1994 / Revised version received August 11, 1995     

A degenerate Neumann problem for quasilinear elliptic integro-differential operators

This paper is devoted to the study of the following degenerate Neumann problem for a quasilinear elliptic integro-differential operator Here is a second-order elliptic integro-differential operator of Waldenfels type and is a first-order Ventcel' operator with a(x) and b(x) being non-negative smooth functions on such that on . Classical existence and uniqueness results in the framework of H?lder spaces are derived under suitable regularity and structure conditions on the nonlinear term f(x,u,Du). Received April 22, 1997; in final form March 16, 1998     

Four versus two solutions of semilinear elliptic boundary value problems

This paper concerns the existence of four (or six) solutions of semilinear elliptic boundary value problems provided that two disorderly solutions are known. The results are obtained under very generic conditions. Received: 26 August 2000 / Accepted: 23 February 2001 / Published online: 23 July 2001     

The boundary integro-differential equations of three-dimensional Neumann problem in linear elasticity

Summary. In this paper, we mainly consider the three dimensional Neumann problem in linear elasticity, which is reduced to a system of integro-differential equations on the boundary based on a new representation of the derivatives of the double-layer potential. Furthermore a new boundary finite element method for this Neumann problem is presented. Received July 13, 1992 / Revised version received June 28, 1993     

A posteriori error estimates for optimal control problems governed by parabolic equations

Summary. In this paper, we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation. We obtain a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive finite element approximation schemes for the control problem. Received July 7, 2000 / Revised version received January 22, 2001 / Published online January 30, 2002 RID="*" ID="*" Supported by EPSRC research grant GR/R31980     

An efficient and accurate parallel algorithm for the singular value problem of bidiagonal matrices

Summary. In this paper we propose an algorithm based on Laguerre's iteration, rank two divide-and-conquer technique and a hybrid strategy for computing singular values of bidiagonal matrices. The algorithm is fully parallel in nature and evaluates singular values to tiny relative error if necessary. It is competitive with QR algorithm in serial mode in speed and advantageous in computing partial singular values. Error analysis and numerical results are presented. Received March 15, 1993 / Revised version received June 7, 1994     

On the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and Dirichlet boundary conditions

We give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in bounded domains with Dirichlet boundary conditions. We impose mild conditions on the domains and lower order (nonlinear) coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by Kato class functions. Our approach uses an implicit probabilistic representation, Schauder's fixed point theorem, and new a priori estimates for solutions of the corresponding linear elliptic equations. In the course of deriving these a priori estimates we show that the Green functions for operators of the form on D are comparable when one modifies the drift term b on a compact subset of D. This generalizes a previous result of Ancona [2], obtained under an condition on b, to a Kato condition on . Received: 21 April 1998 / in final form 26 March 1999     

Wavelet solutions for the Dirichlet problem

Summary. A modified classical penalty method for solving a Dirichlet boundary value problem is presented. This new fictitious domain penalty method eliminates the traditional need of generating a complex computation grid in the case of irregular domains. It is based on the fact that one can expand the boundary measure under the chosen basis which leads to a fast, approximate calculation of boundary integral. The compact support and orthonormality of the basis are essential for representing the boundary measure numerically, and therefore for implementing this methodology. Received June 3, 1992 / Revised version received November 8, 1993     

Exit time moments, boundary value problems, and the geometry of domains in Euclidean space

Let X _t be a diffusion in Euclidean space. We initiate a study of the geometry of smoothly bounded domains in Euclidean space using the moments of the exit time for particles driven by X _t , as functionals on the space of smoothly bounded domains. We provide a characterization of critical points for each functional in terms of an overdetermined boundary value problem. For Brownian motion we prove that, for each functional, the boundary value problem which characterizes critical points admits solutions if and only if the critical point is a ball, and that all critical points are maxima. Received: 23 January 1997 / Revised version: 21 January 1998     

A posteriori error estimation for fully discrete hierarchic models of elliptic boundary value problems on thin domains

Summary. A posteriori error estimators for fully discrete hierarchic modelling on thin domains are derived and are shown to provide computable upper bounds on the discretization error and on the total error. The estimators are shown to be robust and do not degenerate as the thickness of the domain tends to zero. If the discretization part of the error is negligible, the estimator for the modelling error reduces to the one recently obtained for semi-discrete hierarchical modelling by Babuska and Schwab. Received July 25, 1996 / Revised version received July 31, 1997     

An inverse problem for an elliptic equation with an affine term

We consider the following elliptic boundary value problem: on , u = 0 on where is a smooth bounded planar domain. We show that for a large class of domains and for any such that is not identically constant there exist at most finitely many different pairs of coefficients such that the problem has a solution with the normal flux on . Received: 4 February 1999     

Stable parallel elimination for boundary value odes

Summary. A parallelizable and vectorizable algorithm for solving linear algebraic systems arising from two-point boundary value ODEs is described. The method is equivalent to Gaussian elimination, with row partial pivoting, applied to a certain column-reordered version of the usual almost-block-diagonal coefficient matrix. We present analytical and numerical evidence to show that the algorithm is stable in most circumstances. Results from implementation on a shared-memory multiprocessor and a vector processor are given. The approach can be extended to handle problems with multipoint and integral conditions or with algebraic parameters. Received April 26, 1991 / Revised version received September 3, 1993     

On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations

Summary. For the numerical solution of (non-necessarily well-posed) linear equations in Banach spaces we consider a class of iterative methods which contains well-known methods like the Richardson iteration, if the associated resolvent operator fulfils a condition with respect to a sector. It is the purpose of this paper to show that for given noisy right-hand side the discrepancy principle (being a stopping rule for the iteration methods belonging to the mentioned class) defines a regularization method, and convergence rates are proved under additional smoothness conditions on the initial error. This extends similar results obtained for positive semidefinite problems in Hilbert spaces. Then we consider a class of parametric methods which under the same resolvent condition contains the method of the abstract Cauchy problem, and (under a weaker resolvent condition) the iterated method of Lavrentiev. A modified discrepancy principle is formulated for them, and finally numerical illustrations are presented. Received August 29, 1994 / Revised version received September 19, 1995     

Effective preconditioning of Uzawa type schemes for a generalized Stokes problem

Summary. The Schur complement of a model problem is considered as a preconditioner for the Uzawa type schemes for the generalized Stokes problem (the Stokes problem with the additional term in the motion equation). The implementation of the preconditioned method requires for each iteration only one extra solution of the Poisson equation with Neumann boundary conditions. For a wide class of 2D and 3D domains a theorem on its convergence is proved. In particular, it is established that the method converges with a rate that is bounded by some constant independent of . Some finite difference and finite element methods are discussed. Numerical results for finite difference MAC scheme are provided. Received May 2, 1997 / Revised version received May 10, 1999 / Published online May 8, 2000