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.

On decoupled time step/subcycling and iteration strategies for multiphysics problems

“…In this case, the meshes and discretization methods will usually also be different, so some care has to be taken here (e.g. see Carey et al [19] for an example involving solution of a shallow-water subsystem on a finite element mesh with projection of the velocity field (in one-way coupling) to a different mesh using control volume approximation for transport simulation). We mention the relevance to grid computing, but conclude by noting that grid latency is the key factor here.…”

Constrained interpolation (remap) of divergence-free fields

“…Prolongation and restriction operators in multilevel and adaptive methods, methods on non-matching grids, and computer simulations of problems with multiple physics and/or scales are just few examples that require this capability; see, for example, [2], [8], [10], [15], [19], [22], and the references cited therein.…”

“…Many of the existing algorithms also impose additional restrictions, such as grid hierarchy, or factor-of-two refinement. Interpolation on unstructured grids is considered in [8] and [10]. However, these papers adopt the technique of Lagrange multipliers to enforce the relevant constraint.…”

A Hybrid Spectral/Finite-Volume Algorithm for Large-Eddy Simulation of Scalars in the Atmospheric Boundary Layer

“…It is essential that the interpolated velocity field preserves the divergence-free character locally on the new grid to high precision (Carey et al 2001).…”

“…In Carey et al (2001), a finite-element solution of the velocity field is coupled with a coarser finite-volume water quality model. They proposed a solution based on Lagrangian multipliers to enforce the local divergence-free constraint on the finitevolume grid.…”