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FEM for singularly perturbed stationary drift-diffusion equations with boundary layers

by R. Hiptmair and T. Yu

(Report number 2025-16)

Abstract
This work investigates a finite element method (FEM) based on the weak imposition of Dirichlet boundary conditions using Nitsche's method for solving singularly perturbed stationary drift-diffusion equations in the quasineutral limit. A priori error analysis is conducted in the underresolved regime, where the mesh is too coarse to effectively resolve boundary layers, showing optimal convergence rates for the proposed method. Numerical experiments additionally demonstrate improved accuracy in terms of global and interior discretization errors as well as contact current evaluation. Further insights into the observed ``plateaus'' in the convergence plots are obtained by studying a one-dimensional reaction-diffusion problem.discretization errors as well as contact current evaluation. Further insights into the observed “plateaus” in the convergence plots are obtained by studying a one-dimensional reaction-diffusion problem.

Keywords: drift-diffusion equations, singular perturbation, boundary layers, weakly imposed boundary conditions

@Techreport{HY25_1137,
  author = {R. Hiptmair and T. Yu},
  title = {FEM for singularly perturbed stationary drift-diffusion equations with boundary layers},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2025-16},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2025/2025-16.pdf },
  year = {2025}
}

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