Study of the discontinuous Galerkin methods and their a posteriori error estimates

间断Galerkin方法作为与有限元方法密切相关的数值方法，近年来引起了人们的广泛关注。我们对这些方法进行了理论分析，开发了新的方法，改进了一些方法，并将它们应用于一些具体问题。此外，我们还列出了这类方法的后验估计所需的一些误差常数，并得到了其中一些常数的具体数值或上界。为了验证理论结果的有效性和寻找新的研究对象，我们进行了数值实验，并对所考虑的方法的可能性进行了一些深入的了解。特别是，间断Galerkin方法在近似函数和单元形状的选择上具有灵活性，因此有望改进现有的有限元方法。

The discontinuous Galerkin methods have recently drawn much attention as numerical methods closely related to the finite element methods. We have performed theoretical analysis of such methods, developed new methods, improved some methods, and applied them to some concrete problems. Moreover, we have listed up some error constants required for a posteriori estimates of such methods, and obtained concrete values or upper bounds for some of such constants. Numerical experiments were performed to see the validity of theoretical results and to find new research subjects, and we obtained some insights to the possibility of the considered methods. In particular, the discontinuous Galerkin methods are flexible in the selection of approximate functions and element shapes, so that they are expected to improve the existing finite element methods.