Adaptive finite elements for nonlinear transport equations



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The a posteriori error analysis and estimation for conforming finite element approximation of stationary boundary value problems exhibiting certain classes of nonlinear reaction and nonlinear diffusion was investigated. Principal contributions were: (C1) Derivation of new rational local error indicators for both spatial and parameter error in parameterized nonlinear reaction–diffusion problems, (C2) New continuation algorithms for turning point prediction and calculation using adaptive mesh refinement (AMR), (C3) Improved linearization theory for nonlinear diffusion systems, (C4) A posteriori error analysis and new local error indicators for global error and error in output functionals for nonlinear diffusion systems, and (C5) A study of nonlinear diffusive mass transport in a PEM fuel cell cathode using AMR. For parameterized nonlinear reaction–diffusion problems, the solutions to a pair of local linear boundary value problems on each element were postprocessed to create local and global error indicators for both the spatial and parameter error, which were tested on representative problems, including the catalyst pellet problem from chemical engineering. The estimation of critical parameter values at simple turning points was demonstrated using AMR and the new local error indicator for the parameter error. The linearization theory for nondifferentiable, nonlinear diffusion operators with nonlinear solution–dependent diffusion coefficients was extended to systems, including the Stefan–Maxwell multicomponent diffusion operator. In addition, the application of the linearization arguments to the a posteriori error analysis of these operators was justified. Local error indicators for global error and error in output functionals were derived, based on solving local linear boundary value problems that approximate the primal and dual error. Numerical studies demonstrated the performance of the new indicators and confirmed the advantages of the linearization approach over simple estimates of the residuals. Finally, a study of nonlinear diffusive mass transport in the cathode of a PEM fuel cell was conducted, illustrating the use of AMR and the new local error indicators in an application problem of general interest. Calculation of an effectiveness factor that measures mass transport limitations in the cathode was also explored.