TY - JOUR
T1 - Interior Estimates of Semidiscrete Finite Element Methods for Parabolic Problems with Distributional Data
AU - Guo , Li
AU - Li , Hengguang
AU - Yang , Yang
JO - Journal of Computational Mathematics
VL - 4
SP - 458
EP - 474
PY - 2019
DA - 2019/02
SN - 37
DO - http://doi.org/10.4208/jcm.1804-m2017-0240
UR - https://global-sci.org/intro/article_detail/jcm/13002.html
KW - Parabolic problems, Distributional data, Finite element methods, Interior estimates, Well-posedness, Singularity.
AB - <p style="text-align: justify;">Let $Ω ⊂ \mathbb{R}^d$, $1 ≤ d ≤ 3$, be a bounded $d$-polytope. Consider the parabolic equation
on $Ω$ 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 of
the equation. In particular, we derive that the interior error is bounded by the best local
approximation error, the negative norms of the error, and the negative norms of the time
derivative of the error. This result implies different convergence rates for the numerical
solution in different interior regions, especially when the region is close to the singular
point. Numerical test results are reported to support the theoretical prediction.</p>