Volume 16, Issue 2
A Discontinuous Ritz Method for a Class of Calculus of Variations Problems

Int. J. Numer. Anal. Mod., 16 (2019), pp. 340-356.

Published online: 2018-10

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• Abstract

This paper develops an analogue (or counterpart) to discontinuous Galerkin (DG) methods for approximating a general class of calculus of variations problems. The proposed method, called the discontinuous Ritz (DR) method, constructs a numerical solution by minimizing a discrete energy over DG function spaces. The discrete energy includes standard penalization terms as well as the DG finite element (DG-FE) numerical derivatives developed recently by Feng, Lewis, and Neilan in [7]. It is proved that the proposed DR method converges and that the DG-FE numerical derivatives exhibit a compactness property which is desirable and crucial for applying the proposed DR method to problems with more complex energy functionals. Numerical tests are provided on the classical $p$-Laplace problem to gauge the performance of the proposed DR method.

• Keywords

Variational problems minimizers discontinuous Galerkin (DG) methods DG finite element numerical calculus compactness convergence.

65N30 65N12 35J60.

xfeng@math.utk.edu (Xiaobing Feng)

sschnake@ou.edu (Stefan Schnake)

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@Article{IJNAM-16-340, author = {Feng , Xiaobing and Schnake , Stefan}, title = {A Discontinuous Ritz Method for a Class of Calculus of Variations Problems}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2018}, volume = {16}, number = {2}, pages = {340--356}, abstract = {

This paper develops an analogue (or counterpart) to discontinuous Galerkin (DG) methods for approximating a general class of calculus of variations problems. The proposed method, called the discontinuous Ritz (DR) method, constructs a numerical solution by minimizing a discrete energy over DG function spaces. The discrete energy includes standard penalization terms as well as the DG finite element (DG-FE) numerical derivatives developed recently by Feng, Lewis, and Neilan in [7]. It is proved that the proposed DR method converges and that the DG-FE numerical derivatives exhibit a compactness property which is desirable and crucial for applying the proposed DR method to problems with more complex energy functionals. Numerical tests are provided on the classical $p$-Laplace problem to gauge the performance of the proposed DR method.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/12807.html} }
TY - JOUR T1 - A Discontinuous Ritz Method for a Class of Calculus of Variations Problems AU - Feng , Xiaobing AU - Schnake , Stefan JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 340 EP - 356 PY - 2018 DA - 2018/10 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/12807.html KW - Variational problems KW - minimizers KW - discontinuous Galerkin (DG) methods KW - DG finite element numerical calculus KW - compactness KW - convergence. AB -

This paper develops an analogue (or counterpart) to discontinuous Galerkin (DG) methods for approximating a general class of calculus of variations problems. The proposed method, called the discontinuous Ritz (DR) method, constructs a numerical solution by minimizing a discrete energy over DG function spaces. The discrete energy includes standard penalization terms as well as the DG finite element (DG-FE) numerical derivatives developed recently by Feng, Lewis, and Neilan in [7]. It is proved that the proposed DR method converges and that the DG-FE numerical derivatives exhibit a compactness property which is desirable and crucial for applying the proposed DR method to problems with more complex energy functionals. Numerical tests are provided on the classical $p$-Laplace problem to gauge the performance of the proposed DR method.

Xiaobing Feng & Stefan Schnake. (2020). A Discontinuous Ritz Method for a Class of Calculus of Variations Problems. International Journal of Numerical Analysis and Modeling. 16 (2). 340-356. doi:
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