In this paper, we present a multigrid preconditioner for solving the linear system arising from the piecewise linear nonconforming CrouzeixRaviart discretization of second order elliptic problems with jump coefficients. The preconditioner uses the standard conforming subspaces as coarse spaces. Numerical tests show both robustness with respect to the jump in the coefficient and near-optimality ...

This paper deals with a finite element method to solve interior fluid-structure vibration problems valid for compressible and incompressible fluids. It is based on a displacement formulation for both the fluid and the solid. The pressure of the fluid is also used as a variable for the theoretical analysis yielding a well posed mixed linear eigenvalue problem. Lowest order triangular Raviart-Tho...

Computer algebra systems often have to deal with piecewise continuous functions. These are, for example, the absolute value function, signum, piecewise defined functions but also functions that are the supremum or infimum of two functions. We present a new algebraic approach to these types of problems. This paper presents a normal form for a function ring containing piecewise polynomial functio...

Abstract In this paper a new method to compute approximate analytical solutions for differential equations of fractional order is presented. The proposed computational method, called the Piecewise Polynomial Least Squares Method (PWPLSM), combination and piecewise-defined functions. Numerical results different orders are discussed. obtained with PWPLSM compared other existing numerical solution...

Approaches for the identification of piecewise affine systems are presented. The identification problem for piecewise affine systems is formulated as a smooth constrained, and a nonsmooth unconstrained optimization problem. The main advantage of the new problem formulations is the fact that no mixed integer problems have to be solved within this method. The feasibility and performance of the pr...

A hierarchy of one-dimensional high-order remapping schemes is presented and their performance with respect to accuracy and convergence rate investigated. The schemes are also compared based on remapping experiments in closed domains. The piecewise quartic method (PQM) is presented, based on fifth-order accurate piecewise polynomials, and is motivated by the need to significantly improve hybrid...

For decades researchers have struggled with the problem of envy-free cake cutting: how to divide a divisible good between multiple agents so that each agent likes his own allocation best. Although an envy-free cake cutting protocol was ultimately devised, it is unbounded, in the sense that the number of operations can be arbitrarily large, depending on the preferences of the agents. We ask whet...

We establish lower bounds on the complexity of computing the following number-theoretic functions and relations from piecewise linear primitives: (i) the Legendre and Jacobi symbols, (ii) pseudoprimality, and (iii) modular exponentiation. As a corollary to the lower bound obtained for (i), an algorithm of Shallit and Sorenson is optimal (up to a multiplicative constant).