Error Estimation Using Numerical Smoothing Indicator
by
Champike Attanayake
Bowling Green State University, Bowling Green OH, 43403

We all know Lax-Richtmyer Theorem:

Consistency + Numerical Stability ⇒ Convergence.

In practice, it is hard if not impossible, to verify numerical stability of a scheme while solving an evolution equation, especially if the equation is nonlinear and/or the scheme is complex. A large gap exists between error analysis theory and numerical computation practice. We propose that one can use numerical smoothing to replace numerical stability in error estimate. As a property of a (computed) numerical solution, the numerical smoothing property can be monitored by a "smoothing indicator". This approach has several advantages such as, it works for nonlinear systems and for any time stepping scheme (as long as the smoothing indicator remains bounded). Hence, we can narrow the gap between theory and practice significantly. In this talk, I focus on a finite element solution of parabolic problems.

Date received: September 28, 2007