In this paper we analyze a homogeneous parabolic problem with initial data in the space of regular Borel measures. The problem is discretized in time with a discontinuous Galerkin scheme of arbitrary degree and in space with continuous finite elements of orders one or two. We show parabolic smoothing results for the continuous, semidiscrete and fully discrete problems. Our main results are interior $L^\infty$ error estimates for the evaluation at the endtime, in cases where the initial data is supported in a subdomain. In order to obtain these, we additionally show interior $L^\infty$ error estimates for $L^2$ initial data and quadratic finite elements, which extends the corresponding result previously established by the authors for linear finite elements.

翻译：本文研究具有正则Borel测度空间初值的齐次抛物问题。时间方向采用任意次数的间断Galerkin格式离散，空间方向使用一阶或二阶连续有限元。我们给出了连续问题、半离散问题及全离散问题的抛物光滑性结果。主要结论为：当初值支集位于子区域时，对终时刻评估建立了内部$L^\infty$误差估计。为获得上述结果，我们进一步证明了$L^2$初值与二次有限元情形下的内部$L^\infty$误差估计，这扩展了作者此前针对线性有限元建立的相应结论。