Papers by Bernard Bialecki
Journal of Integral Equations and Applications, Sep 1, 1992
Siam Journal on Numerical Analysis, 2001
We study the orthogonal spline collocation (OSC) solution of a homogeneous Dirichlet boundary val... more We study the orthogonal spline collocation (OSC) solution of a homogeneous Dirichlet boundary value problem in a rectangle for a general nonlinear elliptic partial differential equation. The approximate solution is sought in the space of Hermite bicubic splines. We prove local existence and uniqueness of the OSC solution, obtain optimal order H 1 and H 2 error estimates, and prove the quadratic convergence of Newton's method for solving the OSC problem.
Siam Journal on Numerical Analysis, Jul 26, 2006
Siam Journal on Numerical Analysis, 2004
Numerische Mathematik, Jun 30, 1997
Multilevel preconditioners are proposed for the iterative solution of the discrete problems which... more Multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet boundary value problem for a self-adjoint elliptic partial differential equation on a rectangle. Additive and multiplicative preconditioners are defined respectively as sums and products of independent operators on a sequence of nested subspaces of the fine partition approximation space. A general theory of additive and multiplicative Schwarz methods is used to prove that the preconditioners are spectrally equivalent to the collocation discretization of the Laplacian with the spectral constants independent of the fine partition stepsize and the number of levels. The preconditioned conjugate gradient and preconditioned Orthomin methods are considered for the solution of collocation problems. An implementation of the methods is discussed and the results of numerical experiments are presented. Subject Classification (1991): 65F10, 65N30
Siam Journal on Numerical Analysis, Sep 8, 1999
SIAM Journal on Numerical Analysis, 2000
We study the orthogonal spline collocation (OSC) solution of a homogeneous Dirichlet boundary val... more We study the orthogonal spline collocation (OSC) solution of a homogeneous Dirichlet boundary value problem in a rectangle for a general nonlinear elliptic partial differential equation. The approximate solution is sought in the space of Hermite bicubic splines. We prove local existence and uniqueness of the OSC solution, obtain optimal order H 1 and H 2 error estimates, and prove the quadratic convergence of Newton's method for solving the OSC problem.
SIAM Journal on Numerical Analysis, 2003
SIAM Journal on Numerical Analysis, 2003
SIAM Journal on Numerical Analysis, 2009
SIAM Journal on Numerical Analysis, 2008
New numerical techniques are presented for the solution of a class of linear partial integro-diff... more New numerical techniques are presented for the solution of a class of linear partial integro-differential equations (PIDEs) with a positive-type memory term in the unit square. In these methods, orthogonal spline collocation (OSC) is used for the spatial discretization, and, for the time stepping, new alternating direction implicit (ADI) methods based on the backward Euler, the Crank-Nicolson, and the second order BDF methods combined with judiciously chosen quadrature rules are considered. The ADI OSC methods are proved to be of optimal accuracy in time and in the L 2 norm in space. Numerical results confirm the predicted convergence rates and also exhibit optimal accuracy in the L ∞ and H 1 norms and superconvergence phenomena.
SIAM Journal on Numerical Analysis, 1998
... aP+q+Svn/OXPOyqatS is continuous in QT, for 0 < p < i, O0 q &lt... more ... aP+q+Svn/OXPOyqatS is continuous in QT, for 0 < p < i, O0 q < j, O < s < k, and n - 1, 2. If X is a normed space with norm II Ix, then we define LP(X), p= 2, oo, by LP(X)- {V: V(. , t) EX, t E [0, T], IIVIILP(x) < oo}, where IVIIL2(X) ( IIVI2xdt , VILo(X) ess supo<t<TlV] lx. (JO( TII II ...
Numerische Mathematik, 1997
Multilevel preconditioners are proposed for the iterative solution of the discrete problems which... more Multilevel preconditioners are proposed for the iterative solution of the discrete problems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet boundary value problem for a self-adjoint elliptic partial differential equation on a rectangle. Additive and multiplicative preconditioners are defined respectively as sums and products of independent operators on a sequence of nested subspaces of the fine partition approximation space. A general theory of additive and multiplicative Schwarz methods is used to prove that the preconditioners are spectrally equivalent to the collocation discretization of the Laplacian with the spectral constants independent of the fine partition stepsize and the number of levels. The preconditioned conjugate gradient and preconditioned Orthomin methods are considered for the solution of collocation problems. An implementation of the methods is discussed and the results of numerical experiments are presented. Subject Classification (1991): 65F10, 65N30
Numerical Algorithms, 2009
Abstract Matrix decomposition algorithms (MDAs) employing fast Fourier transforms are developed f... more Abstract Matrix decomposition algorithms (MDAs) employing fast Fourier transforms are developed for the solution of the systems of linear algebraic equations arising when the finite element Galerkin method with piecewise Hermite bicubics is used to solve Poisson's ...
Molecular Physics, 2000
A basis of Hermite splines is used in conjunction with the collocation method to solve the orbita... more A basis of Hermite splines is used in conjunction with the collocation method to solve the orbital equations for diatomic molecules. Accurate solutions of the Hartree-Fock equations are obtained using iterative methods over most regions of space, while solving the equations by Gaussian elimination near the nuclear centres. In order to improve the speed and accuracy of our iterative scheme,
Mathematics of Computation, 1993
A complete stability and convergence analysis is given for two-and three-level, piecewise Hermite... more A complete stability and convergence analysis is given for two-and three-level, piecewise Hermite bicubic orthogonal spline collocation, Laplacemodified and alternating-direction schemes for the approximate solution of linear parabolic problems on rectangles. It is shown that the schemes are unconditionally stable and of optimal-order accuracy in space and time.
Journal of Physics B: Atomic, Molecular and Optical Physics, 1996
A basis of Hermite splines is used in conjunction with the collocation method to obtain accurate ... more A basis of Hermite splines is used in conjunction with the collocation method to obtain accurate solutions of the Schrödinger equation for 0953-4075/29/12/006/img7. The spectral transform variation of the Lanczos method is found to be the most suitable method for solving the matrix eigenvalue problem, while the preconditioned conjugate gradient method is very effective for solving the systems of linear
Journal of Numerical Mathematics, 2000
A nonoverlapping domain decomposition approach with uniform and matching grids is used to define ... more A nonoverlapping domain decomposition approach with uniform and matching grids is used to define and compute the orthogonal spline collocation solution of the Dirichlet boundary value problem for Poisson's equation on a square partitioned into four squares. The collocation solution on four interfaces is computed using the preconditioned conjugate gradient method with the preconditioner defined in terms of interface preconditioners for the adjacent squares. The collocation solution on four squares is computed by a matrix decomposition method that uses fast Fourier transforms. With the number of preconditioned conjugate gradient iterations proportional to log 2 N , the total cost of the algorithm is O(N 2 log 2 N ), where the number of unknowns in the collocation solution is O(N 2 ). The approach presented in this paper, along with that in [6], generalizes to variable coefficient equations on rectangular polygons partitioned into many subrectangles and is well suited for parallel computation.
Journal of Integral Equations and Applications, 1992
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Papers by Bernard Bialecki