# Error Estimation of the Besse Relaxation Scheme for a Semilinear Heat Equation

## TL;DR

This paper provides the first error estimate for a fully discrete Besse relaxation scheme applied to a semilinear heat equation, demonstrating optimal second-order accuracy in a combined temporal and spatial norm.

## Contribution

It introduces a new stability argument and establishes the first error bounds for the fully discrete Besse relaxation scheme in this context.

## Key findings

- Achieved optimal second-order error estimate in discrete $L_t^{ar{ ext{infinity}}}(H_x^1)$-norm.
- Developed a novel composite stability argument.
- First literature report of error estimates for fully discrete Besse relaxation scheme.

## Abstract

The solution to the initial and Dirichlet boundary value problem for a semilinear, one dimensional heat equation is approximated by a numerical method that combines the Besse relaxation scheme in time (C. R. Acad. Sci. Paris S{\'e}r. I, vol. 326 (1998)) with a central finite difference method in space. A new, composite stability argument is developed, leading to an optimal, second-order error estimate in the discrete $L_t^{\infty}(H_x^1)-$norm. It is the first time in the literature where an error estimate for fully discrete approximations based on the Besse relaxation scheme is provided.