Locally constrained projections on grids

Abstract

A technique is formulated for projecting vector fields from one unstructured computational grid to another grid so that a constraint condition such as a conservation property holds at the cell or element level on the ‘receiving’ grid. The approach is based on ideas from constrained optimization and certain mixed or multiplier-type finite element methods in which Lagrange multipliers are introduced on the elements to enforce the constraint. A theoretical analysis and estimates for the associated saddle-point problem are developed and a new algorithm is proposed for efficient solution of the resulting discretized problem. In the algorithm a reduced Schur's complement problem is constructed for the multipliers and the projected velocity computation reduces to a post-processing calculation. In some instances the reduced system matrix can be factored so that repeated projections involve little more than forward and backward substitution sweeps. Numerical tests with an element of practical interest demonstrate optimal rate of convergence for the projected velocities and verify the local conservation property to expected machine precision. A practical demonstration for environmental simulation of Florida Bay concludes the study. Copyright © 2001 John Wiley & Sons, Ltd.