Discrete least‐squares global approximations to solutions of partial differential equations

Abstract

Solutions of partial differential equations (PDEs) using globally nonvanishing approximating functions are discussed, and the particular case of global polynomial solutions is studied. Convergence and error bounds are examined. Examples are given and compared with analytic solutions. This method seems particularly well suited for elliptic PDEs with continuous boundary conditions and nonhomogeneous terms, even for irregular domains, offering geometric convergence rates. By providing the minimized residues, strong error indicators are obtained. This algorithm's implementation retains simplicity under a variety of applications.

Discrete weighted least-squares method for the Poisson and biharmonic problems on domains with smooth boundary

“…It means that instead of searching for the Lebesgue points for a domain Ω, we have to search for matching points ensuring a compromise between two requirements. Firstly, we want to compute the most accurate approximation of (1)- (2). Secondly, to achieve the highest possible accuracy we need the coefficient matrix A with condition number as small as possible.…”

“…Particularly, it is well suited for the numerical solution of problems governed by partial differential equations. Although the articles concerning the numerical solution of partial differential equations using its pure discrete form are very rare in the literature [1,2], the continuous form in connection with the finite element approximation is a very attractive method for the solution of many boundary value problems using direct and mixed formulations [3]. This acceptance is evident in spite of two nontrivial bottlenecks of the least-squares approach.…”