UNIVERSITY OF COLORADO AT DENVER

PLACE: Mathematics Conference Room 626 UCD Building, 1250 14th St., Denver

TIME: noon (Refreshments served at 11:45 am)

DATE: October 8, Thursday, 1998

Orthogonal Spline Collocation for Nonlinear Dirichlet Problems

Abdulrakhim Aitbayev and Bernard Bialecki

Department of Mathematical and Computer Sciences
Colorado School of Mines

We study the orthogonal spline collocation (OSC) solution  of a
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.  In this work, we prove
existence and uniqueness of the OSC solution, derive optimal order
H^1 and H^2 error estimates, prove the quadratic convergence of
Newton's method for solving the OSC problem, prove convergence of the
preconditioned conjugate gradient (PCG) method for the solution of a
linearized problem at each Newton's iteration, derive a preconditioning
technique for the PCG method based on a matrix decomposition algorithm,
and present numerical results that support the theoretical analysis.

The iterative scheme that we apply for solving the nonlinear algebraic
equations can be described as a double stage Newton-PCG method.  At
each Newton's iteration, we solve, using the PCG method, a linear
system with a symmetric positive definite matrix that corresponds to
the ``normal equation'' of the linearized OSC problem. As a
preconditioner, we choose a matrix corresponding to a special form
of a separable partial differential operator.  We show that the
convergence rate of the PCG method is independent of h.

To some degree, this work fills the gap existing at the theoretical
level between spline collocation and finite element Galerkin methods
for solving nonlinear elliptic boundary value problems.