Residual-Balanced Exponentially Fitted Finite Volumes on Adaptive Equidistributed Meshes for Singularly Perturbed Convection–Diffusion Problems

Abstract

An exponentially fitted finite-volume discretization is coupled to a residual-curvature equidistribution law for one-dimensional convection–diffusion–reaction problems with unresolved outflow layers. The cell flux is obtained by exact local integration and expressed through the Bernoulli function, while the mesh monitor combines a recovered curvature term with an analytically integrated exponential-layer proxy. The resulting tridiagonal operator is an irreducible 𝑀-matrix for positive reaction and convection, independently of the local Péclet number. A discrete comparison argument yields a parameter-explicit maximum-norm estimate, and the equidistribution relation converts it to a parameter-uniform first-order bound up to the monitor regularization error. Computations for 10−8 ≤ 𝜀 ≤ 10−2 show that the worst-case maximum error decreases from 8.9831 × 10−2 at 𝑁 = 32 to 4.5284 × 10−3 at 𝑁 = 512, whereas uniform central and uniform fitted meshes remain unresolved. The adaptive fitted scheme preserves nonpositive off-diagonal coefficients over the full parameter sweep and outperforms a first-order Shishkin upwind method from 𝑁 = 128 onward.