Abstract
The PCD (piecewise constant distributions) method is a discretization technique of the boundary value problems in which the unknown distribution and its derivatives are represented by piecewise constant distributions but on distinct meshes. It has the advantage of producing the most sparse stiffness matrix resulting from the approximate problem. In this contribution, we propose a general PCD triangulation by combining rectangular elements and triangular elements. We also apply this discretization technique for the elasticity problem. We end with presentation of numerical results of the proposed method for the 2D diffusion equation.
Highlights
Formulation of the problemTo keep the presentation of this discretization as simple as possible, we restrict the present contribution to the analysis of the 2D diffusion equation on a nonuniform rectangular mesh
It leads to the most sparse stiffness matrix resulting from the approximate problem
The main issue of the present work is the presentation of a BVP discretization method on polygonal domain. It is based on the use of piecewise constant distributions to represent the unknown distribution as well as its derivatives on distinct meshes
Summary
To keep the presentation of this discretization as simple as possible, we restrict the present contribution to the analysis of the 2D diffusion equation on a nonuniform rectangular mesh. The convergence analysis and the technical results of the PCD method can be found in [9] and [12]. The extension of the presented method on polygonal or L-shaped domain does not raise any difficulties. Where n denotes the unit normal to Γ = ∂Ω and Γ = Γ0 ∪ Γ1. The discrete version of this problem will be based on its variational formulation: find u ∈ H such that ∀ v ∈ H a( u, v ) = (s, v)Ω where H = HΓ10(Ω) = {v ∈ H1(Ω), v = 0 on Γ0},. (s, v)Ω denotes the L2(Ω) scalar product
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