We propose and analyze a space-time Local Discontinuous Galerkin method for the approximation of the solution to parabolic problems. The method allows for very general discrete spaces and prismatic space-time meshes. Existence and uniqueness of a discrete solution are shown by means of an inf-sup condition, whose proof does not rely on polynomial inverse estimates. Moreover, for piecewise polynomial spaces satisfying an additional mild condition, we show a second inf-sup condition that provides an additional control of the time derivative of the discrete solution. We derive hp-a priori error bounds based on these inf-sup conditions, which we use to prove convergence rates for standard, tensor-product, and quasi-Trefftz polynomial spaces. Numerical experiments validate our theoretical results.

翻译：本文提出并分析了一种用于逼近抛物问题解的时空局部间断Galerkin方法。该方法允许使用非常一般的离散空间与棱柱形时空网格。通过满足inf-sup条件证明了离散解的存在唯一性，其证明过程不依赖于多项式逆估计。此外，对于满足附加温和条件的分片多项式空间，我们证明了第二个inf-sup条件，该条件提供了对离散解时间导数的额外控制。基于这些inf-sup条件，我们推导了hp先验误差界，并利用其证明了标准多项式空间、张量积多项式空间以及拟Trefftz多项式空间的收敛速率。数值实验验证了我们的理论结果。