Interior estimates of semidiscrete finite element methods for parabolic problems with distributional data

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Department of Mathematical Sciences


Let Ω ⊂ R" role="presentation" style="box-sizing: border-box; display: inline-block; line-height: 0; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; color: rgb(44, 44, 44); font-family: Helvetica, Tahoma, Arial, PingFangSC-Regular, "Hiragino Sans GB", "Heiti SC", STXihei, "Microsoft YaHei", SimHei, "WenQuanYi Micro Hei"; position: relative;">RRd, 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.

Publication Title

Journal of Computational Mathematics