Sparse grid discontinuous Galerkin methods for high-dimensional elliptic equations

JOURNAL OF COMPUTATIONAL PHYSICS

卷:314

Article

Wang, Zixuan¹  Tang, Qi²  Guo, Wei¹  Cheng, Yingda¹ 

[1] Michigan State Univ, Dept Math, E Lansing, MI 48824 USA

[2] Rensselaer Polytech Inst, Dept Math Sci, Troy, NY 12180 USA

关键词: Discontinuous Galerkin methods;    Interior penalty methods;    Sparse grid;    High-dimensional partial differential;    equations;   

DOI  :  10.1016/j.jcp.2016.03.005

来源: Elsevier

【 摘 要 】

This paper constitutes our initial effort in developing sparse grid discontinuous Galerkin (DG) methods for high-dimensional partial differential equations (PDEs). Over the past few decades, DG methods have gained popularity in many applications due to their distinctive features. However, they are often deemed too costly because of the large degrees of freedom of the approximation space, which are the main bottleneck for simulations in high dimensions. In this paper, we develop sparse grid DG methods for elliptic equations with the aim of breaking the curse of dimensionality. Using a hierarchical basis representation, we construct a sparse finite element approximation space, reducing the degrees of freedom from the standard O(h(-d)) to O(h(-1)vertical bar log(2)h vertical bar d(-1)) for d-dimensional problems, where his the uniform mesh size in each dimension. Our method, based on the interior penalty (IP) DG framework, can achieve accuracy of O(h(k)vertical bar log(2)h vertical bar d(-1)) in the energy norm, where k is the degree of polynomials used. Error estimates are provided and confirmed by numerical tests in multi-dimensions. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free