Equidistributed error mesh for problems with exponential boundary layers

Abstract

In this paper we present a new piecewise-linear finite element mesh suitable for the discretization of the one-dimensional convection–diffusion equation - ε u ″ - bu ′ = 0 , u ( 0 ) = 0 , u ( 1 ) = 1 . The solution to this equation exhibits an exponential boundary layer which occurs also in more complicated convection–diffusion problems of the form - ε Δ u - b ∂ u / ∂ x + cu = f . The new mesh is based on the equidistribution of the interpolation error and it takes into account finite computer arithmetic. It is demonstrated numerically that for the above problem, the new mesh has remarkably better convergence properties than the well-known Shishkin and Bakhvalov meshes.