An extended Galerkin analysis for elliptic problems
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https://doi.org/10.1007/s11425-019-1809-7
An extended Galerkin analysis for elliptic problems Qingguo Hong1 , Shuonan Wu2 & Jinchao Xu1,∗ 1Department
of Mathematics, Pennsylvania State University, University Park, PA 16802, USA; of Mathematical Sciences, Peking University, Beijing 100871, China
2School
Email: [email protected], [email protected], [email protected] Received August 22, 2019; accepted November 21, 2020
Abstract
A general analysis framework is presented in this paper for many different types of finite ele-
ment methods (including various discontinuous Galerkin methods).
For the second-order elliptic equation −div(α∇u) = f , this framework employs four different discretization variables, uh , ph , u ˇh and pˇh , where uh and ph are for approximation of u and p = −α∇u inside each element, and u ˇh and pˇh are for approximation of residual of u and p · n on the boundary of each element. The resulting 4-field discretization is proved to satisfy two types of inf-sup conditions that are uniform with respect to all discretization and penalization parameters. As a result, many existing finite element and discontinuous Galerkin methods can be analyzed using this general framework by making appropriate choices of discretization spaces and penalization parameters. Keywords MSC(2010)
finite element method, extended Galerkin analysis, unified study 65N30, 65M60, 65M12
Citation: Hong Q G, Wu S N, Xu J. An extended Galerkin analysis for elliptic problems. Sci China Math, 2021, 64, https://doi.org/10.1007/s11425-019-1809-7
1
Introduction
In this paper, we propose an extended Galerkin analysis framework for most of the existing finite element methods (FEMs). We will illustrate the main idea by using the following elliptic boundary value problem: − div(α∇u) = f in Ω, (1.1) u=0 on ΓD , − (α∇u) · n = 0 on ΓN , where Ω ⊂ Rd (d > 1) is a bounded domain and its boundary, ∂Ω, is split into Dirichlet and Neumann parts, namely ∂Ω = ΓD ∪ ΓN . For simplicity, we assume that the (d − 1)-dimensional measure of ΓD is nonzero. Here, n is the outward unit normal direction of ΓN , and α : Rd → Rd is a bounded and symmetric positive definite matrix, with its inverse denoted by c = α−1 . By setting p = −α∇u, the above problem can be written as { cp + ∇u = 0 in Ω, (1.2) − divp = −f in Ω * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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2
Sci China Math
with the boundary condition u = 0 on ΓD and p · n = 0 on ΓN . There are two major variational formulations for (1.1). The first is to find 1 u ∈ HD (Ω) := {v ∈ H 1 (Ω) : v |ΓD = 0} 1 such that for any v ∈ HD (Ω),
∫
∫ (α∇u) · ∇v dx = Ω
f v dx.
(1.3)
Ω
The second one is to find p ∈ HN (div; Ω) := {q ∈ H(div; Ω) : q · n = 0 on ΓN },
u ∈ L2 (Ω)
such that for any q ∈ HN (div; Ω) and v ∈ L2 (Ω), ∫ ∫ cp · q dx − u divq dx = 0, Ω Ω ∫ ∫ − v divp dx = − f v dx. Ω
(1.4)
Ω
In correspondence to the two variational formulations, two diff
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