A Discrete Comparison Principle Analysis of an Improved Scheme for Time‐Fractional Weakly Singular Mixed Parabolic–Elliptic Models Driven by Fusion Welding Processes

ABSTRACT Fusion welding is an empirical approach that involves uniting materials by melting their interfaces using an external heat source and subsequent solidification of the adjacent regions. The mathematical modeling of welding processes is notably challenging owing to the associated complex physical phenomena. The present study aims to analyze an efficient numerical method for a class of time‐fractional mixed parabolic–elliptic models arising in fusion welding processes. The considered domain is partitioned into two subdomains, the first of which is characterized by a time‐fractional parabolic reaction–diffusion problem. The second one poses a steady state convection–diffusion–reaction problem of the elliptic type. The presence of weak singularities near leads to a layer in the solution, and this adversely affects the performance of polynomial interpolation discretization schemes, leading to a loss of convergence on uniform meshes. To achieve an optimal convergence rate, the fractional derivative is discretized using the L1 scheme on a layer‐resolving graded mesh, while a class of second‐order finite difference schemes is employed in the spatial direction. The discrete comparison principle and a suitable choice of the barrier function are used to establish the convergence analysis. Numerical experiments and comparisons with the literature verify the efficiency and applicability of the proposed method.