Convergence properties of a class of boundary element approximations to linear diffusion problems with localized nonlinear reactions

Anthony P. Peirce, Attila Askar, Herschel Rabitz

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We consider a boundary element (BE) Algorithm for solving linear diffusion desorption problems with localized nonlinear reactions. The proposed BE algorithm provides an elegant representation of the effect of localized nonlinear reactions, which enables the effects of arbitrarily oriented defect structures to be incorporated into BE models without having to perform severe mesh deformations. We propose a one‐step recursion procedure to advance the BE solution of linear diffusion localized nonlinear reaction problems and investigate its convergence properties. The separation of the linear and nonlinear effects by the boundary integral formulation enables us to consider the convergence properties of approximations to the linear terms and nonlinear terms of the boundary integral equation separately. For the linear terms we investigate how the degree of piecewise polynomial collocation in space and the size of the spatial mesh relative to the time step affects the accumulation of errors in the one‐step recursion scheme. We develop a novel convergence analysis that combines asymptotic methods with Lax's Equivalence Theorem. We identify a dimensionless meshing parameter θ whose magnitudé governs the performance of the one‐step BE schemes. In particular, we show that piecewise constant (PWC) and piecewise linear (PWL) BE schemes are conditionally convergent, have lower asymptotic bounds placed on the size of time steps, and which display excess numerical diffusion when small time steps are used. There is no asymptotic bound on how large the tie steps can be–this allows the solution to be advanced in fewer, larger time steps. The piecewise quadratic (PWQ) BE scheme is shown to be unconditionally convergent; there is no asymptotic restriction on the relative sizes of the time and spatial meshing and no numerical diffusion. We verify the theoretical convergence properties in numerical examples. This analysis provides useful information about the appropriate degree of spatial piecewise polynomial and the meshing strategy for a given problem. For the nonlinear terms we investigate the convergence of an explicit algorithm to advance the solution at an active site forward in time by means of Caratheodory iteration combined with piecewise linear interpolation. We consider a model problem comprising a singular nonlinear Volterra equation that represents the effect of the term in the BE formulation that is due to a single defect. We prove the convergence of the piecewise linear Caratheodory iteration algorithm to a solution of the model problem for as long as such a solution can be shown to exist. This analysis provides a theoretical justification for the use of piecewise linear Caratheodory iterates for advancing the effects of localized reactions.

Original languageEnglish (US)
Pages (from-to)75-108
Number of pages34
JournalNumerical Methods for Partial Differential Equations
Issue number1
StatePublished - 1990

All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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