A posteriori error estimation with finite element semi- and fully discrete methods for nonlinear parabolic equations in one space dimension.(English)Zbl 0798.65089

The paper considers $$p$$-th order finite element discretization of $\left[ u_ t = \bigl[ a(u)u_ x \bigr]_ x - f(u);\quad u \mid_{x = c,d} = 0 \right]$ giving error $$\approx Ch^ p$$. After computation (or concurrently for adaptive schemes) local estimates of $$C$$ can be computed where needed by solving any of several variant approximating problems for the next order correction. Under reasonable assumptions (smooth solution, uniform ellipticity, etc.), these are proved to give asymptotically accurate estimates. Experimental results show one already gets estimates within a few percent for moderate $$h$$ even in more general cases where these theoretical results do not apply.

MSC:

 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35K55 Nonlinear parabolic equations
Full Text: