Hybrid Spectral Difference Methods for an Elliptic Equation

Abstract

A locally conservative, hybrid spectral difference method (HSD) is presented and analyzed for the Poisson equation. The HSD is composed of two types of finite difference approximations; the cell finite difference and the interface finite difference. Embedded static condensation on cell interior unknowns considerably reduces the global couplings, resulting in the system of equations in the cell interface unknowns only. A complete ellipticity analysis is provided. The optimal order of convergence in the semi-discrete energy norms is proved. Several numerical results are given to show the performance of the method, which confirm our theoretical findings.

Hybrid Spectral Difference Methods for Elliptic Equations on Exterior Domains with the Discrete Radial Absorbing Boundary Condition

“…For a detailed derivation of the high order finite difference formulas for h and ∂ h ν we refer to [11,12]. Let us define the Gaussian quadratures on a reference cell R (with |R| = h × k) Table 4 Reduction in degrees of freedom by static condensation…”

“…Some numerical analysis of (4.5) can be found in [11], where analysis is performed for the Poisson equation. The hybrid difference method is understood as the finite difference version of the hybridized finite element method.…”

“…The mass conservation property holds in each cell and flux continuity holds across inter-cell boundaries. Embedded static condensation property considerably reduces global degrees of freedom [10][11][12].…”

Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions

“…The immersed hybrid difference method was developed by the first author [14] for the elliptic interface problems, and this idea is extended to solve boundary value problems. In the immersed boundary approach, instead of solving the equation on a boundary fitting mesh, we consider the extended problem of it on the whole domain by treating the inner boundary as an immersed interface.…”

“…The hybrid difference (HD) method is a finite difference version of the hybridized discontinuous Galerkin method and it was introduced by the authors and their colleagues for the elliptic, Stokes and Navier-Stokes equations [15,16,17,18]. Recently, the immersed boundary approach was applied to the HD method to develop the IHD method by the first author [14] for elliptic interface problems. The key feature of the IHD method is the introduction of the virtual to real transformation (VR-T) to find the finite difference stencil on interface cells, which refer to the cells that contain a portion of the given interface.…”