Modified nodal cubic spline collocation for biharmonic equations

Abstract

We formulate a modified nodal cubic spline collocation scheme for the solution of the biharmonic Dirichlet problem on the unit square. We prove existence and uniqueness of a solution of the scheme and show how the scheme can be solved on an N × N uniform partition of the square at a cost O(N 2 log 2 N + mN 2 ) using fast Fourier transforms and m iterations of the preconditioned conjugate gradient method. We demonstrate numerically that m proportional to log 2 N guarantees the desired convergence rates. Numerical results indicate the fourth order accuracy of the approximations in the global maximum norm and the fourth order accuracy of the approximations to the first order partial derivatives at the partition nodes.

Compact optimal quadratic spline collocation methods for the Helmholtz equation

“…The approach that we have adopted has been employed in the formulation of a compact optimal QSC method for the biharmonic Dirichlet problem in [27]; cf., [1]. Its extension to the development of compact QSC methods for Helmholtz problems in other coordinate systems (cf., [14,15]) and for parabolic problems in one and two space variables (cf., [17,18]) are topics of future research.…”

Matrix decomposition algorithms for elliptic boundary value problems: a survey

“…(It appears that there is no corresponding analytical formula for U (2) for the finite difference scheme of [190].) Using (3.19), (3.20), the fact that z (0) = −6/ h, and properties of…”

“…The method has the advantage that, in contrast to the OSMs of [31,32], step 2 of the MDA involves the solution of tridiagonal systems instead of pentadiagonal systems. This scheme is used in [2] to approximate the biharmonic Dirichlet problem comprising (3.23) and (3.26), and the resulting linear system is solved using the Schur complement approach and an MDA. This algorithm costs O (N 2 log N + mN 2 ) with m iterations of the preconditioned conjugate gradient method.…”

Optimal superconvergent one step quadratic spline collocation methods

“…Recently this problem was solved in [1] using optimal C 2 cubic spline collocation. Future research will involve extending the new QSC methods for second-order elliptic problems to non-rectangular regions using domain decomposition.…”