Interior Estimates of Semidiscrete Finite Element Methods for Parabolic Problems with Distributional Data (Journal of Computational Mathematics, 2019)
Let $Ω ⊂ \mathbb{R}^d$, $1 ≤ d ≤ 3$, be a bounded $d$-polytope. Consider the parabolic equationon $Ω$ with the Dirac delta function on the right hand side. We study the well-posedness,regularity, and the interior error estimate of semidiscrete finite element approximations ofthe equation. In particular, we derive that the interior error is bounded by the best localapproximation error, the negative norms of the error, and the negative norms of the timederivative of the error. This result implies different convergence rates for the numericalsolution in different interior regions, especially when the region is close to the singularpoint. Numerical test results are reported to support the theoretical prediction.
Keywords: Parabolic problems; Distributional data; Finite element methods; Interior estimates; Well-posedness; Singularity.