2000, Vol 1, No 1 http://hdl.handle.net/123456789/3342 2019-06-16T10:32:19Z Algorithm with guaranteed accuracy for computing a solution to an initial value problem for linear difference equations http://hdl.handle.net/123456789/3402 Algorithm with guaranteed accuracy for computing a solution to an initial value problem for linear difference equations Vaskevich, Vladimir; Bulgak, Haydar; Çinar, Cengiz Consider an initial value problem for simultaneous linear difference equations x(n+1) = Ax(n) + f(n), x(0) = a, withA theNN rational matrix, {f(n)} a sequence ofN-dimensional rational vectors, anda a rationalN-dimensional vector. The problem has a unique solution; but to compute {x(n)} in the interval [0,M] withM a nonnegative integer, we approximate the reals and carry out the elementary arithmetic operations in special way. By means of this algorithm, we solve the initial value problem for a discrete asymptotically stable matrixA with guaranteed accuracy. 2000-01-01T00:00:00Z Adaptive hierarchical tenzor product finite elements for fluid dynamics http://hdl.handle.net/123456789/3400 Adaptive hierarchical tenzor product finite elements for fluid dynamics Zenger, Christoph; Schneider, S. A. H. This article is a contribution to the current research field of computational fluid dynamics. We discretize the Stokes flow for Re=0 with adaptive hierarchical finite elements and verify the method with numerical results for the three-dimensional lid-driven cavity problem. In order to solve the corresponding Stokes problem, we replace the constraint of the conservation of mass by an elliptic boundary value problem for the pressure distribution p. Consequently, the solution of the Stokes problem is reduced to the solution of d +1 Poisson problems, the so-called successive poisson scheme. We use the hierarchical tensor product finite element method for the numerical solution of the Poisson problems as a basic module. On one hand, this allows a straightforward approach for the self-adaptive solution process: We start with a regular discretization and create new elements, where the hierarchical surplus of the weak divergence indicates the need to refine. On the other hand, we use multigrid concepts for the efficient solution of the large linear systems arising from the elliptic differential equations. The discussed example shows that the use of elements with variable aspect ratio pays off for the resolution of line singularities. 2000-01-01T00:00:00Z The Cauchy problem for pseudohyperbolic equations http://hdl.handle.net/123456789/3346 The Cauchy problem for pseudohyperbolic equations Demidenko, Gennadii In this paper we study the Cauchy problem for a class of partial differential equations not solved with respect to the highest derivative. The members of this class are called pseudohyperbolic equations. We establish the solvability conditions, obtain an ``energy" inequality for solutions, and prove the well-posedness of the Cauchy problem in the weighted Sobolev spaces. 2000-01-01T00:00:00Z Factorized sparse approximate inverses for preconditioning and smoothing http://hdl.handle.net/123456789/3345 Factorized sparse approximate inverses for preconditioning and smoothing Huckle, Thomas In recent papers the use of sparse approximate inverses for the preconditioning of linear equations Ax=b is examined. The minimization of || AM-I || in the Frobenius norm generates good preconditioners without any a priori knowledge on the pattern of M. For symmetric positive definite A and a given a priori pattern there exist methods for computing factorized sparse approximate inverses L with LLT A-1. Here, we want to modify these algorithms that they are able to capture automatically a promising pattern for L. We use these approximate inverses for solving linear equations with the cg-method. Furthermore we introduce and test modifications of this method for computing factorized sparse approximate inverses that are suited for smoothing in multigrid solvers. 2000-01-01T00:00:00Z