A Sequential Least Squares Method for Poisson Equation Using a Patch Reconstructed Space

Abstract

We propose a new least squares finite element method to solve the Poisson equation. By using a piecewisely irrotational space to approximate the flux, we split the classical method into two sequential steps. The first step gives the approximation of flux in the new approximation space and the second step can use flexible approaches to give the pressure. The new approximation space for flux is constructed by patch reconstruction with one unknown per element consisting of piecewisely irrotational polynomials. The error estimates in the energy norm and L 2 norm are derived for the flux and the pressure. Numerical results verify the convergence order in error estimates, and demonstrate the flexibility and particularly the great efficiency of our method.

Reconstructed Discontinuous Approximation to Stokes Equation in a Sequential Least Squares Formulation

“…In this section, we propose a sequential least squares finite element method to solve the Stokes problem based on the first-order system (2.4). We are motivated by the idea in [16,27] to decouple the system (2.4) into two steps. The first first-order system is defined to seek the numerical approximations to the gradient U and the pressure p, which reads…”

“…The novelty is that we construct three specific approximation spaces which allow us to solve the Stokes problem in two sequential steps. The sequential process is motivated from the idea in [16,27] to define two least-squares-type functionals to approximate unknowns sequentially. The feasibility of this method is based on new approximation spaces which are obtained by solving local least squares problems on each element.…”

Reconstructed Discontinuous Approximation to Stokes Equation in A Sequential Least Squares Formulation

“…In this section, we define three types of reconstruction operators that will be used in numerically solving (2.4). The first one is the reconstruction operator which has been used in [24,22,25] and the other two operators are extensions from the first one. The reconstruction procedure includes two parts.…”

“…We refer to [9] and the references therein for an overview of least squares finite element methods. Based on discontinuous approximation, the discontinuous least squares finite element methods have also been developed for many problems including the Stokes problem, and we refer to [7,6,4,3,25] for more details.…”

A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction

“…The discontinuous Galerkin method by patch reconstruction (PRDG) was firstly introduced in [12] by Li et al to solve elliptic equations. The method has been successfully applied into the biharmonic problem [11], the Stokes problem [13,14], the eigenvalue problem [15], the linear elasticity problem [16], the convection-diffusion problem [19] and the sequential least square method [17]. This method was originally motivated by the patch reconstruction technique in the finite volume scheme for the hydrodynamic solver [20].…”