Fully Discrete Pointwise Smoothing Error Estimates for Measure Valued Initial Data
Provides rigorous error bounds for numerical solutions of parabolic PDEs with singular initial data, which is important for applications involving point sources or concentrated loads.
The paper proves interior L∞ error estimates for fully discrete approximations of parabolic problems with measure-valued initial data, using discontinuous Galerkin time-stepping and continuous finite elements of orders one or two. The results extend previous work to quadratic finite elements and include smoothing estimates for the continuous, semidiscrete, and fully discrete problems.
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.