A Comparison of Additive Schwarz Preconditioners for Parallel Adaptive Finite Elements

Abstract

We consider a second order elliptic boundary value problem in the variational form: find u ^∗ ∈ H ₀ ¹(Ω), for a given polygonal (polyhedral) domain $\varOmega \subset \mathbb{R}^{d},\,d = 2,3$ and a source term f ∈ L ²(Ω), such that

$$\displaystyle{ \underbrace{\mathop{\int _{\varOmega }\nabla u^{{\ast}}(x) \cdot \nabla v(x)\,dx}}\limits _{\equiv a(u^{{\ast}},v)} =\underbrace{\mathop{ \int _{\varOmega }f(x)v(x)\,dx}}\limits _{\equiv (f,v)},\quad \text{for all }v \in H_{0}^{1}(\varOmega ). }$$

(1)

The Bank–Holst parallel adaptive meshing paradigm [1–3] is utilised to solve (1) in a combination of domain decomposition and adaptivity.