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Numerical Null-Solutions to Iterated Dirac Operator on Bounded Domains

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Abstract

The main purpose of this paper is to study numerical null-solutions to the iterated Dirac operator on bounded domains by using methods of discrete Clifford analysis. First, we study the properties of discrete Euler operators, introduce its inverse operators, and construct a discrete version of the Almansi-type decomposition theorem for the iterated discrete Dirac operator. Then, we give representations of numerical null-solutions to the iterated Dirac operator on a bounded domain in terms of its Taylor series. Finally, in order to illustrate our numerical approach, we present a simple numerical example in form of a discrete approximation of the Stokes’ equation, and show its convergence to the corresponding continuous problem when the lattice constant goes to zero.

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Acknowledgments

The authors cordially thank two anonymous referees for their valuable comments which lead to the improvement of this paper.

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Authors and Affiliations

  1. Department of Mathematics, CIDMA, University of Aveiro, Aveiro, Portugal

    Min Ku & Uwe Kähler

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  1. Min Ku
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  2. Uwe Kähler
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Correspondence to Min Ku.

Additional information

Communicated by Irene Sabadini.

This work was supported in part by the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e a Tecnologia”) through the CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013, and by the Postdoctoral Foundation from FCT (Portugal) under Grant No. SFRH/BPD/74581/2010.

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Ku, M., Kähler, U. Numerical Null-Solutions to Iterated Dirac Operator on Bounded Domains. Complex Anal. Oper. Theory 11, 307–328 (2017). https://doi.org/10.1007/s11785-016-0544-5

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