Finite difference method

Abstract: In problems with interfaces, the unknown or its derivatives may have jump discontinuities. Finite difference methods, including the method of A. Mayo and the immersed interface method of R. LeVeque and Z. Li, maintain accuracy by adding corrections, found from the jumps, to the difference operator at grid points near the interface and by modifying the operator if necessary. It has long been observed that the solution can be computed with uniform O.h 2 / accuracy even if the truncation error is O.h/ at the interface, while O.h 2 / in the interior. We prove this fact for a class of static interface problems of elliptic type using discrete analogues of estimates for elliptic equations. Moreover, we show that the gradient is uniformly accurate to O.h 2 log.1=h//. Various implications are discussed, including the accuracy of these methods for steady fluid flow governed by the Stokes equations. Two-fluid problems can be handled by first solving an integral equation for an unknown jump. Numerical examples are presented which confirm the analytical conclusions, although the observed error in the gradient is O.h 2 /.

Abstract: Continuous Media.- The Finite Difference Method.- The Finite Element Method.- Elements of Numerical Algorithms.- Phase Transformations.- Deformation of Solids.- Incompressible Fluid Flow.- Inverse Methods.- Stochastic Methods.- Appendices.

Abstract: We exploit the mimetic finite difference method introduced by Hyman and Shashkov to present a framework for estimating vector fields and related scalar fields (divergence, curl) of physical interest from image sequences. Our approach provides a basis for consistent definitions of higher-order differential operators, for the analysis and a novel stability result concerning second-order div-curl regularizers, for novel variational schemes to the estimation of solenoidal (divergence-free) image flows, and to convergent numerical methods in terms of subspace corrections.