Discontinuous Galerkin isogeometric analysis for elliptic problems with discontinuous diffusion coefficients on surfaces
- PDF / 839,533 Bytes
- 20 Pages / 439.642 x 666.49 pts Page_size
- 54 Downloads / 231 Views
Discontinuous Galerkin isogeometric analysis for elliptic problems with discontinuous diffusion coefficients on surfaces Stephen Edward Moore1 Received: 9 April 2019 / Accepted: 14 November 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper is concerned with using discontinuous Galerkin isogeometric analysis (dG-IGA) as a numerical treatment of diffusion problems on orientable surfaces ⊂ R3 . The computational domain or surface considered consists of several nonoverlapping subdomains or patches which are coupled via an interior penalty scheme. In Langer and Moore [13], we presented a priori error estimate for conforming computational domains with matching meshes across patch interface and a constant diffusion coefficient. However, in this article, we generalize the a priori error estimate to non-matching meshes and discontinuous diffusion coefficients across patch interfaces commonly occurring in industry. We construct B-spline or NURBS approximation spaces which are discontinuous across patch interfaces. We present a priori error estimate for the symmetric discontinuous Galerkin scheme and numerical experiments to confirm the theory. Keywords Discontinuous Galerkin · Multi-patch isogeometric analysis · Elliptic problems · a priori error analysis · Surface PDE · Interior penalty Galerkin · Laplace-Beltrami · Discontinuous coefficients
1 Introduction In this paper, we consider the second-order elliptic boundary value problem on an open, smooth, connected, and oriented two-dimensional surface ⊂ R3 as follows: find u : → R such that − div (α∇ u) + u = f in ,
u = 0 on D ,
n · (α∇ u) = gN on N , (1.1)
Stephen Edward Moore
[email protected] 1
Department of Mathematics, University of Cape Coast, Cape Coast, Ghana
Numerical Algorithms
where the diffusion coefficient α is uniformly bounded, i.e., αmin ≤ α ≤ αmax with positive constants αmax and αmin , f and gN are given sufficiently smooth data. The physical or computational domain ⊂ R3 is a compact, connected, and positively oriented surface with boundary ∂. The boundary of the computational domain consists of the Dirichlet part D with positive boundary measure and a Neumann part N such that ∂ := D N . The operators div and ∇ are the surface divergence and surface gradient, respectively, and will be defined in Section 2. Partial differential equations (PDEs) on surfaces arise in many fields of application like material science, fluid mechanics, electromagnetics, biology, and image processing, see, e.g., [7] for several interesting discussions on applications. For several years, numerical methods dedicated to the solutions of PDEs on manifolds including conforming and non-conforming finite element methods (FEM) have been well studied and applied to compute the solution of elliptic and parabolic evolution problems on fixed and evolving computational domains, see, e.g., [5, 7]. We note that there are however some drawbacks to the standard surface FEM. The standard surface FEM has two main sou
Data Loading...
