"Two-Dimensional Singularly Perturbed Problems: Boundary Layers, Interior Layers and Uniform Convergence of Numerical Methods"
Sesión nº. 6 Seminario de Matemática Aplicada 2025/2026
El próximo viernes, 15 de mayo tendrá lugar el sexto seminario del Departamento de Matemática Aplicada en la EINA, correspondiente al Curso 2025/2026.
- Lecturer: Dr. Ram Shiromani
- Conference title: "Two-Dimensional Singularly Perturbed Problems: Boundary Layers, Interior Layers and Uniform Convergence of Numerical Methods"
- Abstract:
This talk illustrates, in an accessible and rigorous way, the compelling need for efficient numerical resolution of singularly perturbed differential equations problems in which a small positive diffusion parameter in (0,1] multiplies the highestorder derivative, fundamentally altering the qualitative and quantitative structure of the solution. These equations arise across a broad spectrum of physical and engineering phenomena: the transport and dispersion of contaminants in fluids or porous media, heat and mass transfer in catalytic converters and near oceanic boundaries, chemical reaction--diffusion--convection processes, oil and gas reservoir simulations, blood flow and drug delivery modelling in the human body, as well as ecological and population-dynamics models. The defining mathematical feature is a multiscale solution structure: regions of extremely rapid variation (large gradients) of width O(ε) or O(ε^(1/2), termed boundary layers or interior layers, coexist with regions of smooth, O(1)-varying behavior. Classical methods on uniform meshes fail to resolve these layers efficiently, producing spurious oscillations unless the mesh spacing is taken of order ε everywhere. This motivates the construction and analysis of ε-uniform numerical methods, which achieve convergence independent of the perturbation parameter, typically through the use of layer-adapted meshes (Shishkin, Bakhvalov) or appropriate stabilization. We present the mathematical framework, illustrate layer phenomena through representative two-dimensional examples, survey engineering applications, and discuss the analytical tools required to prove uniform convergence.
Conferenciante: Dr. Ram Shiromani
Día: Viernes, 15 de mayo de 2026
Hora: 12:00 h
Lugar: Aula 22, Edificio Torres Quevedo de la Escuela de Ingeniería y Arquitectura
