Single and Multi-Interval Legendre τ-Methods in Time for Parabolic Equations

In this paper, we take the parabolic equation with periodic boundary conditions as a model to present a spectral method with the Fourier approximation in spatial and single/multi-interval Legendre Petrov–Galerkin method in time. For the single interval spectral method in time, we obtain the optimal error estimate in L ²-norm. For the multi-interval spectral method in time, the L ²-optimal error estimate is valid in spatial. Numerical results show the efficiency of the methods.     

两个空间变量的拟线性抛物方程初值问题解的正则性

本文研究了两个空间变量的拟线性退缩抛物方程Ｃａｕｃｈｙ问题广义解的正则性问题。推广了文献［１］的结果。     

Time-stepping algorithms for semidiscretized linear parabolic PDEs based on rational approximants with distinct real poles

Time dependent problems in Partial Differential Equations (PDEs) are often solved by the Method Of Lines (MOL). For linear parabolic PDEs, the exact solution of the resulting system of first order Ordinary Differential Equations (ODEs) satisfies a recurrence relation involving the matrix exponential function. In this paper, we consider the development of a fourth order rational approximant to the matrix exponential function possessing real and distinct poles which, consequently, readily admits a partial fraction expansion, thereby allowing the distribution of the work in solving the corresponding linear algebraic systems in essentially Backward Euler-like solves on concurrent processors. The resulting parallel algorithm possesses appropriate stability properties, and is implemented on various parabolic PDEs from the literature including the forced heat equation and the advection-diffusion equation.Dedicated to Professor J. Crank on the occasion of his 80th birthday     

On parallel asynchronous high-order solutions of parabolic PDEs

In achieving significant speed-up on parallel machines, a major obstacle is the overhead associated with synchronizing the concurrent processes. This paper presents high-orderparallel asynchronous schemes, which are schemes that are specifically designed to minimize the associated synchronization overhead of a parallel machine in solving parabolic PDEs. They are asynchronous in the sense that each processor is allowed to advance at its own speed. Thus, these schemes are suitable for single (or multi) user shared memory or (message passing) MIMD multiprocessors. Our approach is demonstrated for the solution of the multidimensional heat equation, of which we present a spatial second-order Parametric Asynchronous Finite-Difference (PAFD) scheme. The well-known synchronous schemes are obtained as its special cases. This is a generalization and expansion of the results in [5] and [7]. The consistency, stability and convergence of this scheme are investigated in detail. Numerical tests show that although PAFD provides the desired order of accuracy, its efficiency is inadequate when performed on each grid point.In an alternative approach that uses domain decomposition, the problem domain is divided among the processors. Each processor computes its subdomain mostly independently, while the PAFD scheme provides the solutions at the subdomains' boundaries. We use high-order finite-difference implicit scheme within each subdomain and determine the values at subdomains' boundaries by the PAFD scheme. Moreover, in order to allow larger time-step, we use remote neighbors' values rather than those of the immediate neighbors. Numerical tests show that this approach provides high efficiency and in the case which uses remote neighbors' values an almost linear speedup is achieved. Schemes similar to the PAFD can be developed for other types of equations [3].This research was supported by the fund for promotion of research at the Technion.     

Semidiscrete Galerkin Approximation for a Linear Stochastic Parabolic Partial Differential Equation Driven by an Additive Noise

We study the semidiscrete Galerkin approximation of a stochastic parabolic partial differential equation forced by an additive space-time noise. The discretization in space is done by a piecewise linear finite element method. The space-time noise is approximated by using the generalized L₂ projection operator. Optimal strong convergence error estimates in the L₂ and norms with respect to the spatial variable are obtained. The proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem. The error estimates are applicable in the multi-dimensional case. AMS subject classification (2000) 65M, 60H15, 65C30, 65M65.Received April 2004. Revised September 2004. Communicated by Anders Szepessy.     

Quasilinear evolutionary equations and continuous interpolation spaces

In this paper we analyze the abstract parabolic evolutionary equations

     

regularizing effect in a nonlinear, degenerate parabolic equation in one space dimension

In this paper we provide and analyze a nonlinear degenerate parabolic equation in one space dimension with the following smoothing property: If the initial data is only uniformly continuous, at positive times, the solution has bounded second derivatives (it belongs to ). We call this surprising phenomenon a regularizing effect. So far, such phenomena have only been observed in uniformly parabolic equations.

     

Smooth solutions to a class of free boundary parabolic problems

We establish existence, uniqueness, and regularity results for solutions to a class of free boundary parabolic problems, including the free boundary heat equation which arises in the so-called ``focusing problem' in the mathematical theory of combustion. Such solutions are proved to be smooth with respect to time for positive , if the data are smooth.

     

Semilinear parabolic equations involving measures and low regularity data

A detailed study of abstract semilinear evolution equations of the form is undertaken, where generates an analytic semigroup and is a Banach space valued measure depending on the solution. Then it is shown that the general theorems apply to a variety of semilinear parabolic boundary value problems involving measures in the interior and on the boundary of the domain. These results extend far beyond the known results in this field. A particularly new feature is the fact that the measures may depend nonlinearly and possibly nonlocally on the solution.