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关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Stochastic differential equations and diffusion processes

Stochastic equations in infinite dimensions

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

We propose and analyze a semi-discrete (in time) scheme and a fully discrete scheme for the Allen-Cahn equation u

 t
 −Δu+ɛ−2
f(u)=0 arising from phase transition in materials science, where ɛ is a small parameter known as an ``interaction length''. The primary goal of this paper is to establish some useful a priori error estimates for the proposed numerical methods, in particular, by focusing on the dependence of the error bounds on ɛ. Optimal order and quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function u

 0
 . In particular, all our error bounds depend on only in some lower polynomial order for small ɛ. The cruxes of the analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of de Mottoni and Schatzman [18, 19] and Chen [12] and to establish a discrete counterpart of it for a linearized Allen-Cahn operator to handle the nonlinear term. Finally, as a nontrivial byproduct, the error estimates are used to establish convergence and rate of convergence of the zero level set of the fully discrete solution to the motion by mean curvature flow and to the generalized motion by mean curvature flow.

Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows

We propose and analyze a semi-discrete and a fully discrete mixed finite element method for the Cahn-Hilliard equation ut + Δ(ɛΔu−ɛ−1f(u)) = 0, where ɛ > 0 is a small parameter. Error estimates which are quasi-optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain. In particular, it is shown that all error bounds depend on ** only in some lower polynomial order for small ɛ. The cruxes of our analysis are to establish stability estimates for the discrete solutions, to use a spectrum estimate result of Alikakos and Fusco [2], and Chen [15] to prove a discrete counterpart of it for a linearized Cahn-Hilliard operator to handle the nonlinear term on a stretched time grid. The ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the Hele-Shaw (Mullins-Sekerka) problem as ɛ → 0 in [29].

Error analysis of a mixed finite element method for the Cahn-Hilliard equation

Micromagnetics is a continuum variational theory describing magnetization patterns in ferromagnetic media. Its multiscale nature due to different inherent spatiotemporal physical and geometric scales, together with nonlocal phenomena and a nonconvex side-constraint, leads to rich behavior and pattern formation. This variety of effects is also the reason for severe problems in analysis, model validation, reductions, and numerics, which have only recently been accessed and are reviewed in this work.

/pdf/recent-developments-in-the-modeling-analysis-and-numerics-of-1xq8itdpz0.pdf

Recent Developments in the Modeling, Analysis, and Numerics of Ferromagnetism

The Landau--Lifshitz--Gilbert equation describes the dynamics of ferromagnetism, where strong nonlinearity and nonconvexity are hard to tackle: so far, existing explicit schemes to approximate weak solutions suffer from severe time-step restrictions. In this paper, we propose an implicit fully discrete scheme and verify unconditional convergence.

Convergence of an Implicit Finite Element Method for the Landau--Lifshitz--Gilbert Equation

We study finite element based space-time discretisations of the incompressible Navier-Stokes equations with noise. In three dimensions, sequences of numerical solutions construct weak martingale solutions for vanishing discretisation parameters. In the two dimensional case, numerical solutions converge to the unique strong solution.

Finite-element-based discretizations of the incompressible Navier–Stokes equations with multiplicative random forcing

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