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arXiv · 1709.04297

A Discontinuous Ritz Method for a Class of Calculus of Variations Problems

Xiaobing Feng·Stefan Schnake
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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 [Feng2013]. 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.

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Xiaobing Feng, Stefan Schnake. 2017-09-13. A Discontinuous Ritz Method for a Class of Calculus of Variations Problems. https://arxiv.org/abs/1709.04297

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