Numerical Approximation of Poisson Problems in Long Domains

Abstract In this paper, we consider the Poisson equation on a “long” domain which is Cartesian product of one-dimensional long interval with ( d − 1)-dimensional domain. The right-hand side assumed to have rank-1 tensor structure. We will present and compare methods construct approximations solution structure computational effort governed by only solving elliptic problems lower-dimensional domains. A zero-th order approximation derived using tools from asymptotic analysis (method 1). resulting an elementary and, hence has fixed error turns out be very close best possible order. This can used as starting guess for derivation higher-order greedy-type method 2). Numerical experiments show that converging towards exact solution. Method 3 based via exponential sums applied discretized differential operators their inverses. It proved converges exponentially respect rank. numerical performance sensitivity these three methods.