TITLE:
A Full Predictor-Corrector Finite Element Method for the One-Dimensional Heat Equation with Time-Dependent Singularities
AUTHORS:
Jake L. Nkeck
KEYWORDS:
Singularities, Finite Element Methods, Heat Equation, Predictor-Corrector Algorithm
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.12 No.4,
April
30,
2024
ABSTRACT: The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.