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A new finite element approach for problems containing small geometric details

Wolfgang Hackbusch, Stefan A. Sauter (1998)

Archivum Mathematicum

In this paper a new finite element approach is presented which allows the discretization of PDEs on domains containing small micro-structures with extremely few degrees of freedom. The applications of these so-called Composite Finite Elements are two-fold. They allow the efficient use of multi-grid methods to problems on complicated domains where, otherwise, it is not possible to obtain very coarse discretizations with standard finite elements. Furthermore, they provide a tool for discrete homogenization...

A new one-step smoothing newton method for second-order cone programming

Jingyong Tang, Guoping He, Li Dong, Liang Fang (2012)

Applications of Mathematics

In this paper, we present a new one-step smoothing Newton method for solving the second-order cone programming (SOCP). Based on a new smoothing function of the well-known Fischer-Burmeister function, the SOCP is approximated by a family of parameterized smooth equations. Our algorithm solves only one system of linear equations and performs only one Armijo-type line search at each iteration. It can start from an arbitrary initial point and does not require the iterative points to be in the sets...

A parallel algorithm for two phase multicomponent contaminant transport

Todd Arbogast, Clint N. Dawson, Mary F. Wheeler (1995)

Applications of Mathematics

We discuss the formulation of a simulator in three spatial dimensions for a multicomponent, two phase (air, water) system of groundwater flow and transport with biodegradation kinetics and wells with multiple screens. The simulator has been developed for parallel, distributed memory, message passing machines. The numerical procedures employed are a fully implicit expanded mixed finite element method for flow and either a characteristics-mixed method or a Godunov method for transport and reactions...

A parallel block Lanczos algorithm and its implementation for the evaluation of some eigenvalues of large sparse symmetric matrices on multicomputers

Mario Guarracino, Francesca Perla, Paolo Zanetti (2006)

International Journal of Applied Mathematics and Computer Science

In the present work we describe HPEC (High Performance Eigenvalues Computation), a parallel software package for the evaluation of some eigenvalues of a large sparse symmetric matrix. It implements an efficient and portable Block Lanczos algorithm for distributed memory multicomputers. HPEC is based on basic linear algebra operations for sparse and dense matrices, some of which have been derived by ScaLAPACK library modules. Numerical experiments have been carried out to evaluate HPEC performance...

A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces

Juan Pablo Agnelli, Eduardo M. Garau, Pedro Morin (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this article we develop a posteriori error estimates for second order linear elliptic problems with point sources in two- and three-dimensional domains. We prove a global upper bound and a local lower bound for the error measured in a weighted Sobolev space. The weight considered is a (positive) power of the distance to the support of the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theory hinges on local approximation properties of either Clément or Scott–Zhang interpolation...

A posteriori error estimates for linear exterior problems via mixed-FEM and DtN mappings

Mauricio A. Barrientos, Gabriel N. Gatica, Matthias Maischak (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we combine the dual-mixed finite element method with a Dirichlet-to-Neumann mapping (given in terms of a boundary integral operator) to solve linear exterior transmission problems in the plane. As a model we consider a second order elliptic equation in divergence form coupled with the Laplace equation in the exterior unbounded region. We show that the resulting mixed variational formulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derive the...

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