Restricted Additive Schwarz Preconditioners with Harmonic Overlap for Symmetric Positive Definite Linear Systems

A restricted additive Schwarz (RAS) preconditioning technique was introduced recently for solving general nonsymmetric sparse linear systems. In this paper, we provide one-level and two-level extensions of RAS for symmetric positive definite problems using the so-called harmonic overlaps (RASHO). Both RAS and RASHO outperform their counterparts of the classical additive Schwarz variants (AS). The design of RASHO is based on a much deeper understanding of the behavior of Schwarz-type methods in overlapping subregions and in the construction of the overlap. In RASHO, the overlap is obtained by extending the nonoverlapping subdomains only in the directions that do not cut the boundaries of other subdomains, and all functions are made harmonic in the overlapping regions. As a result, the subdomain problems in RASHO are smaller than those of AS, and the communication cost is also smaller when implemented on distributed memory computers, since the right-hand sides of discrete harmonic systems are always zero and therefore do not need to be communicated. We also show numerically that RASHO-preconditioned CG takes fewer iterations than the corresponding AS-preconditioned CG. A nearly optimal theory is included for the convergence of RASHO-preconditioned CG for solving elliptic problems discretized with a finite element method

Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property

Review: Ph. G. Ciarlet; The finite element method for elliptic problems

Artykuł nie zawiera streszczeniaThe article contains no abstrac

Discussion on the development of applied mathematics in Poland

W 2012 r. Komisja Zastosowań Matematyki (KZM) Komitetu Matematyki Polskiej Akademii Nauk (KM PAN), której mam zaszczyt przewodniczyć w bieżącej kadencji KM, rozpoczęła dyskusję o rozwoju zastosować matematyki w Polsce. Rozpoczęcie tej dyskusji nastąpiło na otwartym Posiedzeniu KZM, które odbyło się pierwszego dnia Konferencji Zastosowań Matematyki w Zakopanem (4 - 11 września 2012) organizowanej przez prof. Łukasza Stettnera. Współorganizatorem tej konferencji odbywającej się co roku od wielu lat, jest KM PAN.In 2012. the Commission of Applied Mathematics (CAM) of the Committee of Mathematics Polish Academy of Sciences (CM PAS), which I have the honor to preside in the current term, started a discussion on the development of applied mathematics in Poland. Started this discussion took place in an open meeting of CAM, which took place on the first day of the Conference of Applied Mathematics in Zakopane (4 - 11 September 2012), organized by prof. Łukasz Stettner.&nbsp

Effective difference schemes for the heat equation in arbitrary regions

.In this paper the author considers the problem of the heat equation ∂u/∂t−(∂2u/∂x21+∂2u/∂x22)=f(x,t) for x∈Ω and t∈(0,T], u(x,0)=φ(x) for x∈Ω, u(x,t)=0 for x∈∂Ω and t∈[0,T]. He constructs a Crank-Nicolson and an alternating direction difference scheme on a regular mesh with steps hi (i=1,2) and τ. Linear interpolation is used for the approximation of the boundary condition. Besides stability of both schemes error estimates are derived under the condition that the derivatives ∂5u/∂t∂x4i and ∂3u/∂t3 are bounded. These estimates are: maxn∥un−yn∥A≤M(τ2+h3/2)andmaxn∥un−yn∥h≤M(τ2+h2+τh1/2+h5/2/τ). Here h=max(h1,h2), un=u(⋅,nτ), yn is the approximate value of un, ∥u∥2h=(u,u)h, (u,v)h=h1h2∑x∈Ωhu(x)v(x) (Ωh is the set of all mesh points lying in Ω), and ∥u∥2A=(u,Au)h where A is the discrete Laplace operator

To the members of the Commission of Applied Mathematics of the Committee of Mathematics of the Polsh Academy of Sciences

Wielce Szanowni Koledzy. Pragnę, jak w ubiegłym roku, złożyć krótkie sprawozdanie z posiedzenia KZM, które odbyło się w Zakopanem w dniu 7.09.2010 r. w czasie trwania XXXIX Konferencji Zastosowań Matematyki (KM PAN był jej współorganizatorem). Posiedzenie było otwarte, brali w niej również udział uczestnicy konferencji nie będący członkami KZM. Głównym punktem programu posiedzenia była dyskusja: o rozwoju edukacji w zakresie zastosowań matematyki, w szczególności matematyki przemysłowej, na uczelniach polskich, o przyszłości nauk matematycznych w Polsce. Przed dyskusją zaproszone odczyty na tematy wyżej wymienione wygłosili profesorowie W. Okrasiński i A. Jakubowski, członkowie naszej Komisji.The author, the Commission Chair, makes a short report on the activity of the Commission in the last year. He pays a special attention to the last meeting of the Commission held in September, 2010 in Zakopane during the XXXIX-th Conference of Applied Mathematics. Two next contributions in this issue of Professors W.~Okrasiński and A.~Jakubowska are written forms of invited presentations to this meeting