In this work, we establish that discontinuous Galerkin methods are capable of producing reliable approximations for a broad class of nonlinear variational problems. In particular, we demonstrate that these schemes provide essential flexibility by removing inter-element continuity while also guaranteeing convergent approximations in the quasiconvex case. Notably, quasiconvexity is the weakest form of convexity pertinent to elasticity. Furthermore, we show that in the non-convex case discrete minimisers converge to minimisers of the relaxed problem. In this case, the minimisation problem corresponds to the energy defined by the quasiconvex envelope of the original energy. Our approach covers all discontinuous Galerkin formulations known to converge for convex energies. This work addresses an open challenge in the vectorial calculus of variations: developing and rigorously justifying numerical schemes capable of reliably approximating nonlinear energy minimization problems with potentially singular solutions, which are frequently encountered in materials science.

翻译：本文证明了不连续伽辽金方法能够为一大类非线性变分问题提供可靠的数值逼近。特别地，我们论证了这些格式通过消除单元间连续性提供了必要的灵活性，同时在拟凸情形下仍能保证逼近解的收敛性。值得注意的是，拟凸性是弹性理论中相关的最弱凸性形式。此外，我们证明了在非凸情形下，离散极小化子收敛于松弛问题的极小化子。此时，极小化问题对应于由原始能量的拟凸包络所定义的能量泛函。我们的方法涵盖了所有已知对凸能量收敛的不连续伽辽金格式。本研究解决了向量变分法中的一个公开挑战：开发并严格论证能够可靠逼近具有潜在奇异解的非线性能量极小化问题的数值格式，这类问题在材料科学中频繁出现。