So far, we have focused entirely on linear problems with constant or piecewise constant coefficients. As we have seen, the methods and analysis for these cases is relatively complete.
In this chapter we expand the discussion to include more complex problems — in particular, problems with smoothly varying coefficients and genuinely nonlinear problems. As we will see, this introduces new elements that need attention, and the analysis of the methods for such problems is more complex. In fact, we will often not attempt to give a complete analysis but merely outline the key results. However, the extension to strongly nonlinear problems displays many unique features and the power and robustness of the discontinuous Galerkin methods.
- Weak Solution
- Nonlinear Problem
- Discontinuous Galerkin Method
- Discontinuous Solution
- Gibbs Phenomenon
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