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Volume 14, Issue 4-5
A Nodal Sparse Grid Spectral Element Method for Multi-Dimensional Elliptic Partial Differential Equations

Zhijian Rong, Jie Shen & Haijun Yu

Int. J. Numer. Anal. Mod., 14 (2017), pp. 762-783.

Published online: 2017-08

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  • Abstract

We develop a sparse grid spectral element method using nodal bases on Chebyshev-Gauss-Lobatto points for multi-dimensional elliptic equations. Since the quadratures based on sparse grid points do not have the accuracy of a usual Gauss quadrature, we construct the mass and stiffness matrices using a pseudo-spectral approach, which is exact for problems with constant coefficients and uniformly structured grids. Compared with the regular spectral element method, the proposed method has the flexibility of using a much less degree of freedom. In particular, we can use less points on edges to form a much smaller Schur-complement system with better conditioning. Preliminary error estimates and some numerical results are also presented.

  • Keywords

Sparse grid, spectral element method, high-dimensional problem, adaptive method.

  • AMS Subject Headings

65N35, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{IJNAM-14-762, author = {Zhijian Rong, Jie Shen and Haijun Yu}, title = {A Nodal Sparse Grid Spectral Element Method for Multi-Dimensional Elliptic Partial Differential Equations}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2017}, volume = {14}, number = {4-5}, pages = {762--783}, abstract = {

We develop a sparse grid spectral element method using nodal bases on Chebyshev-Gauss-Lobatto points for multi-dimensional elliptic equations. Since the quadratures based on sparse grid points do not have the accuracy of a usual Gauss quadrature, we construct the mass and stiffness matrices using a pseudo-spectral approach, which is exact for problems with constant coefficients and uniformly structured grids. Compared with the regular spectral element method, the proposed method has the flexibility of using a much less degree of freedom. In particular, we can use less points on edges to form a much smaller Schur-complement system with better conditioning. Preliminary error estimates and some numerical results are also presented.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/10060.html} }
TY - JOUR T1 - A Nodal Sparse Grid Spectral Element Method for Multi-Dimensional Elliptic Partial Differential Equations AU - Zhijian Rong, Jie Shen & Haijun Yu JO - International Journal of Numerical Analysis and Modeling VL - 4-5 SP - 762 EP - 783 PY - 2017 DA - 2017/08 SN - 14 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/10060.html KW - Sparse grid, spectral element method, high-dimensional problem, adaptive method. AB -

We develop a sparse grid spectral element method using nodal bases on Chebyshev-Gauss-Lobatto points for multi-dimensional elliptic equations. Since the quadratures based on sparse grid points do not have the accuracy of a usual Gauss quadrature, we construct the mass and stiffness matrices using a pseudo-spectral approach, which is exact for problems with constant coefficients and uniformly structured grids. Compared with the regular spectral element method, the proposed method has the flexibility of using a much less degree of freedom. In particular, we can use less points on edges to form a much smaller Schur-complement system with better conditioning. Preliminary error estimates and some numerical results are also presented.

Zhijian Rong, Jie Shen and Haijun Yu. (2017). A Nodal Sparse Grid Spectral Element Method for Multi-Dimensional Elliptic Partial Differential Equations. International Journal of Numerical Analysis and Modeling. 14 (4-5). 762-783. doi:
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