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A method is developed for solving quasilinear convection diffusion problemsstarting on a coarse mesh where the data and solution-dependent coefficientsare unresolved, the problem is unstable and approximation properties do nothold. The Newton-like iterations of the solver are based on the framework ofregularized pseudo-transient continuation where the proposed time integrator isa variation on the Newmark strategy, designed to introduce controllablenumerical dissipation and to reduce the fluctuation between the iterates in thecoarse mesh regime where the data is rough and the linearized problems arebadly conditioned and possibly indefinite. An algorithm and updated markingstrategy is presented to produce a stable sequence of iterates as boundary andinternal layers in the data are captured by adaptive mesh partitioning. Themethod is suitable for use in an adaptive framework making use of local errorindicators to determine mesh refinement and targeted regularization. Derivationand q-linear local convergence of the method is established, and numericalexamples demonstrate the theory including the predicted rate of convergence ofthe iterations.

An improved method for solving quasilinear convection diffusion problems on a coarse mesh