A Discontinuous Ritz Method for a Class of Calculus of Variations Problems
This work provides a new numerical framework for variational problems, extending DG techniques to a broader class of problems, but the results are primarily theoretical with limited empirical validation.
The paper introduces a discontinuous Ritz (DR) method for solving calculus of variations problems, proving convergence and demonstrating compactness of DG-FE numerical derivatives. Numerical tests on the p-Laplace problem validate the method's performance.
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.