6533b83afe1ef96bd12a7844

RESEARCH PRODUCT

Indicators of Errors for Approximate Solutions of Differential Equations

Olli MaliPekka NeittaanmäkiSergey RepinSergey Repin

subject

PhysicsMathematical optimizationDifferential equationComputabilityApproximate solutionUniversal differential equationDifferential algebraic equationType I and type II errorsNumerical partial differential equationsUniversality (dynamical systems)

description

Error indicators play an important role in mesh-adaptive numerical algorithms, which currently dominate in mathematical and numerical modeling of various models in physics, chemistry, biology, economics, and other sciences. Their goal is to present a comparative measure of errors related to different parts of the computational domain, which could suggest a reasonable way of improving the finite dimensional space used to compute the approximate solution. An “ideal” error indicator must possess several properties: efficiency, computability, and universality. In other words, it must correctly reproduce the distribution of errors, be indeed computable, and be applicable to a wide set of approximations. In practice, it is very difficult to satisfy all these requirements simultaneously so that different error indicators are focused on different aims and stress some properties at the sacrifice of others. We discuss the mathematical origins and algorithmic implementation of the most frequently used error indicators. Our goal is twofold: to discuss the main types of error indicators, which have already gained high popularity in numerical practice, and to suggest a unified conception, which covers practically all methods used in error indication.

yearjournalcountryeditionlanguage
2014-01-01
https://doi.org/10.1007/978-94-007-7581-7_2