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A Numerical Study of the Homogeneous Elliptic Equation with Fractional Boundary Conditions

  • Raytcho Lazarov EMAIL logo and Petr Vabishchevich

Abstract

We consider the homogeneous equation Au = 0, where A is a symmetric and coercive elliptic operator in H1(Ω) with Ω bounded domain in ℝd. The boundary conditions involve fractional power α, 0 < α < 1, of the Steklov spectral operator arising in Dirichlet to Neumann map. For such problems we discuss two different numerical methods: (1) a computational algorithm based on an approximation of the integral representation of the fractional power of the operator and (2) numerical technique involving an auxiliary Cauchy problem for an ultra-parabolic equation and its subsequent approximation by a time stepping technique. For both methods we present numerical experiment for a model two-dimensional problem that demonstrate the accuracy, efficiency, and stability of the algorithms.

Acknowledgements

The authors thank their institutions for the support while working on this project. The work of R. Lazarov was partially supported also by grant NSF-DMS # 1620318, while the work of P. Vabishchevich was supported by the Ministry of Education and Science of the Russian Federation (Agreement # 02.a03.21.0008).

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Received: 2016-9-12
Published Online: 2017-4-28
Published in Print: 2017-4-25

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