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, minimizers, discontinuous Galerkin (DG) methods, DG finite element numerical  calculus, compactness, convergence.
AB - <p style="text-align: justify;">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.</p>