Volume 16, Issue 2

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 [B.Bialecki and M.Dryja, A nonoverlapping domain decomposition method for orthogonal spline collocation problems. SIAM J. Numer. Anal. (2003) 41 , 1709 – 1728.], generalizes to variable coefficient equations on rectangular polygons partitioned into many subrectangles and is well suited for parallel computation.

A mathematical model for the accurate computation of the shape of a three-dimensional axisymmetric gas bubble compressed between two horizontal plates is presented. An explicit form of the interface is given by the equation of minimal surfaces, based on an approximation of slow displacements. Conservation of the volume is guaranteed by the introduction of a multiplier. Contact angles are taken into account. The shape of the bubble is numerically solved with a quasi-Newton method that presents fast convergence properties. Numerical results are presented for various compression rates and contact angles.

In this paper we are interested in the coarse-mesh approximations of a class of second order elliptic operators with rough or rapidly oscillatory coefficients. We intend to provide a smoother elliptic operator which on a coarse mesh behaves like the original operator. Note that there is no requirement on smoothness or periodicity of the coefficients. To simplify the theory and the numerical implementations, we restrict ourselves to the one-dimensional case.

This paper presents a numerical method for aerodynamic shape optimization problems with state constraint. It uses a simultaneous semi-iterative technique to solve the equations arising from the first order necessary optimality conditions. Although there is no theoretical proof of convergence of this algorithm so far, in our numerical experiments the method converges without additional globalization in the design space. A reduced SQP method based preconditioner is used for convergence acceleration. Design examples of drag reduction with constant lift for wing and body of a Supersonic Commercial Transport (SCT) aircraft are included. The overall cost of computation is about 8 forward simulation runs.