Finite element approximation for a delayed generalized Burgers-Huxley equation with weakly singular kernels: Part I Well-posedness, regularity and conforming approximation

In this study, we explore the theoretical and numerical aspects of the generalized Burgers-Huxley equation (a non-linear advection-diffusion-reaction problem) incorporating weakly singular kernels in a d-dimensional domain, where $d\in \{2,3\}$. For the continuous problem, we provide an in-depth discussion on the existence and the uniqueness of weak solution using the Faedo-Galerkin approximation technique. Further, regularity results for the weak solution are derived based on assumptions of smoothness for both the initial data and the external forcing. Using the regularity of the solution, the uniqueness of weak solutions has been established. In terms of numerical approximation, we introduce a semi-discrete scheme using the conforming finite element method (CFEM) for space discretization and derive optimal error estimates. Subsequently, we present a fully discrete approximation scheme that employs backward Euler discretization in time and CFEM in space. A priori error estimates for both the semi-discrete and fully discrete schemes are discussed under minimal regularity assumptions. To validate our theoretical findings, we provide computational results that lend support to the derived conclusions.