Sparse grid discontinuous Galerkin methods for high-dimensional elliptic equations

• Published in: Journal of computational physics 2016-06, Vol.314, p.244-263
• Main Authors: Wang, Zixuan, Tang, Qi, Guo, Wei, Cheng, Yingda
• Format: Article
• Language: English
• Subjects: Approximation; Computer simulation; Degrees of freedom; Discontinuous Galerkin methods; Finite element method; Galerkin methods; High-dimensional partial differential equations; Interior penalty methods; Mathematical analysis; Partial differential equations; Polynomials; Sparse grid
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2016.03.005

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|log2⁡h|d−1) for d-dimensional problems, where h is the uniform mesh size in each dimension. Our method, based on the interior penalty (IP) DG framework, can achieve accuracy of O(hk|log2⁡h|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.