Sparse grid discontinuous Galerkin methods for high-dimensional elliptic equations

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• 作者：Zixuan Wang^a ; ^{wangzix1@msu.edu" class="auth_mail" title="E-mail the corresponding author} ; Qi Tang^b ; ^{tangq3@rpi.edu" class="auth_mail" title="E-mail the corresponding author} ; Wei Guo^a ; ^{wguo@math.msu.edu" class="auth_mail" title="E-mail the corresponding author} ; Yingda Cheng^a ; ^{ycheng@math.msu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Discontinuous Galerkin methods ; Interior penalty methods ; Sparse grid ; High-dimensional partial differential equations
• 刊名：Journal of Computational Physics
• 出版年：2016
• 出版时间：1 June 2016
• 年：2016
• 卷：314
• 期：Complete
• 页码：244-263
• 全文大小：3663 K

文摘

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)$O(h^{-d})$ to O(h⁻¹|log₂⁡h|^d−1)$O(h^{-1}|{\log }_{2}h{|}^{d-1})$ 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(h^k|log₂⁡h|^d−1)$O(h^k|{\log }_{2}h{|}^{d-1})$ in the energy norm, where k is the degree of polynomials used. Error estimates are provided and confirmed by numerical tests in multi-dimensions.