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：本文研究了以正则Borel测度空间中的测度为初始数据的齐次抛物型问题。该问题在时间上采用任意次数的间断Galerkin格式进行离散，在空间上采用一阶或二阶连续有限元进行离散。我们给出了连续问题、半离散问题及全离散问题的抛物型光滑性结果。主要成果是在初始数据支撑于子区域的情形下，针对终时刻评估建立了内部$L^\infty$范数误差估计。为获得这些结果，我们进一步证明了针对$L^2$初值和二次有限元的内部$L^\infty$范数误差估计，这推广了作者此前针对线性有限元建立的相应结论。