Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 36, 2011, 661-676

ESTIMATES FOR DEVIATIONS FROM EXACT SOLUTIONS OF THE CAUCHY PROBLEM FOR MAXWELL'S EQUATIONS

Dirk Pauly, Sergey Repin and Tuomo Rossi

Universität Duisburg-Essen, Fakultät für Mathematik
Universitätsstr. 2, 45117 Essen, Germany, dirk.pauly 'at' uni-due.de

V.A. Steklov Mathematical Institute, St. Petersburg Branch
Fontanka 27, 191011 St.\ Petersburg, Russia; repin 'at' pdmi.ras.ru, and
University of Jyväskylä, Department of Mathematical Information Technology
P.O. Box 35 (Agora), FI-40014 Jyväskylä, Finland; serepin 'at' jyu.fi

University of Jyväskylä, Department of Mathematical Information Technology
P.O. Box 35 (Agora), FI-40014 Jyväskylä, Finland; tuomo.rossi 'at' jyu.fi

Abstract. We consider an initial boundary value problem for Maxwell's equations in the space-time cylinder generated by the time interval [0,T]. For this hyperbolic type system, we derive guaranteed and computable upper bounds of the difference between the exact solution and any pair of vector fields that belongs to the natural admissible energy class. Our analysis is based upon transformations of the canonical integral relation and Gronwall's inequality and generalizes the method suggested in [22] for the wave equation to the case of the Maxwell's equation.

2010 Mathematics Subject Classification: Primary 65N15, 35L15, 78M30.

Key words: Cauchy problem for Maxwell's equations, bounds of deviations from exact solutions, a posteriori estimates of the functional type.

Reference to this article: D. Pauly, S. Repin and T. Rossi: Estimates for deviations from exact solutions of the Cauchy problem for Maxwell's equations. Ann. Acad. Sci. Fenn. Math. 36 (2011), 661-676.

Full document as PDF file

doi:10.5186/aasfm.2011.3641