TY - JOUR
T1 - Sparse grid discontinuous Galerkin methods for high-dimensional elliptic equations
AU - Wang, Zixuan
AU - Tang, Qi
AU - Guo, Wei
AU - Cheng, Yingda
N1 - Publisher Copyright:
© 2016 Elsevier Inc.
PY - 2016/6/1
Y1 - 2016/6/1
N2 - 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|log2h|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|log2h|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.
AB - 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|log2h|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|log2h|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.
KW - Discontinuous Galerkin methods
KW - High-dimensional partial differential equations
KW - Interior penalty methods
KW - Sparse grid
UR - http://www.scopus.com/inward/record.url?scp=84962054011&partnerID=8YFLogxK
U2 - 10.1016/j.jcp.2016.03.005
DO - 10.1016/j.jcp.2016.03.005
M3 - Article
AN - SCOPUS:84962054011
SN - 0021-9991
VL - 314
SP - 244
EP - 263
JO - Journal of Computational Physics
JF - Journal of Computational Physics
ER -