A Posteriori Error Estimates for the Crank–Nicolson Method: Application to Parabolic Partial Differential Equations Subject to a Robin Boundary Condition with Small Randomness

• Published in: Journal of scientific computing Vol. 104; no. 1; p. 7
• Main Authors: Shravani, N., Reddy, G. M. M., Vynnycky, M.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2025; Springer Nature B.V
• Subjects: Algorithms; Boundary conditions; Computational Mathematics and Numerical Analysis; Crank-Nicholson method; Estimates; Expected values; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical functions; Mathematics; Mathematics and Statistics; Numerical analysis; Parabolic differential equations; Parameter uncertainty; Partial differential equations; Perturbation methods; Power series; Random variables; Theoretical; Finite element method; 65M15; Perturbation technique; error analysis; Small random input data; Parabolic PDE; 65M60; Crank–Nicolson scheme
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-025-02912-2

In this article, we obtain residual-based a posteriori error estimates for a linear parabolic partial differential equation which is subject to a Robin boundary condition that contains a small uncertainty. To this end, the perturbation technique is exploited to express the exact random solution in terms of a power series with respect to the uncertainty parameter, whence we obtain deterministic problems. Each problem is then discretized in space by continuous piecewise linear elements, and the Crank–Nicolson scheme is used for time-stepping. Reconstruction techniques are employed to obtain optimal bounds. Numerical investigations are performed that confirm the theoretical findings.