A singularly perturbed convection-diffusion parabolic problem with incompatible boundary/initial data

TL;DR

This paper investigates singularly perturbed convection-diffusion parabolic problems with incompatible boundary and initial data, developing analytical and numerical methods to handle data incompatibilities and establishing error bounds supported by numerical experiments.

Contribution

It introduces singular functions for constant coefficient cases and a mixed analytical/numerical approach for variable coefficients with discontinuous data, advancing solution techniques for incompatible data.

Findings

Singular functions match data incompatibilities and satisfy homogeneous equations.

Numerical method's convergence depends on data compatibility levels.

Numerical results confirm theoretical error bounds.

Abstract

A singularly perturbed parabolic problem of convection-diffusion type with incompatible inflow boundary and initial conditions is examined. In the case of constant coefficients, a set of singular functions are identified which match certain incompatibilities in the data and also satisfy the associated homogenous differential equation. When the convective coefficient only depends on the time variable and the initial/boundary data is discontinuous, then a mixed analytical/numerical approach is taken. In the case of variable coefficients and the zero level of compatibility being satisfied (i.e. continuous boundary/initial data), a numerical method is constructed whose order of convergence is shown to depend on the next level of compatibility being satisfied by the data. Numerical results are presented to support the theoretical error bounds established for both of the approaches examined in the paper.