## Journal of Integral Equations and Applications

### Regularized integral formulation of mixed Dirichlet-Neumann problems

#### Abstract

This paper presents a theoretical discussion as well as novel solution algorithms for problems of scattering on smooth two-dimensional domains under Zaremba boundary conditions, for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the proposed numerical methods, which is provided for the first time in the present contribution, concerns detailed information about the singularity structure of solutions of the Helmholtz operator under boundary conditions of Zaremba type. The new numerical method is based on the use of Green functions and integral equations, and it relies on the Fourier continuation method for regularization of all smooth-domain Zaremba singularities as well as newly derived quadrature rules which give rise to high-order convergence, even around Zaremba singular points. As demonstrated in this paper, the resulting algorithms enjoy high-order convergence, and they can be used to efficiently solve challenging Helmholtz boundary value problems and Laplace eigenvalue problems with high-order accuracy.

#### Article information

Source
J. Integral Equations Applications, Volume 29, Number 4 (2017), 493-529.

Dates
First available in Project Euclid: 10 November 2017

https://projecteuclid.org/euclid.jiea/1510282933

Digital Object Identifier
doi:10.1216/JIE-2017-29-4-493

Mathematical Reviews number (MathSciNet)
MR3722840

Zentralblatt MATH identifier
06841172

#### Citation

Akhmetgaliyev, Eldar; Bruno, Oscar P. Regularized integral formulation of mixed Dirichlet-Neumann problems. J. Integral Equations Applications 29 (2017), no. 4, 493--529. doi:10.1216/JIE-2017-29-4-493. https://projecteuclid.org/euclid.jiea/1510282933

#### References

• E. Akhmetgaliyev, Integral equation methods for singular problems and applications to Laplace eigenvalue problems, Ph.D. dissertation, California Insitute of Technology, Pasadena, CA, 2015.
• E. Akhmetgaliyev, O.P. Bruno and N. Nigam, A boundary integral algorithm for the Laplace Dirichlet-Neumann mixed eigenvalue problem, J. Comp. Phys. 298 (2015), 1–28.
• N. Albin and O.P. Bruno, A spectral FC solver for the compressible Navier-Stokes equations in general domains I: Explicit time-stepping, J. Comp. Phys. 230 (2011), 6248–6270.
• D. Britt, S. Tsynkov and E. Turkel, A high-order numerical method for the Helmholtz equation with nonstandard boundary conditions, SIAM J. Sci. Comp. 35 (2013), A2255–A2292.
• O.P. Bruno and S.K. Lintner, Second-kind integral solvers for TE and TM problems of diffraction by open arcs, Radio Sci. 47 (2012).
• O.P. Bruno and M. Lyon, High-order unconditionally stable FC-AD solvers for general smooth domains I, Basic elements, J. Comp. Phys. 229 (2009), 2009–2033.
• F. Cakoni, G.C. Hsiao and W.L. Wendland, On the boundary integral equation method for a mixed boundary value problem of the biharmonic equation, Compl. Var. Th. Appl. 50 (2005), 681–696.
• D.-C. Chang, N. Habal and B.-W. Schulze, The edge algebra structure of the Zaremba problem, J. Pseudo-Differ. Oper. Appl. 5 (2014), 69–155.
• D.L. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Springer, New York, 1998.
• R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in arbitrary $2d$-sectors, Georgian Math. J. 20 (2013), 439–467.
• ––––, Mixed boundary value problems for the Laplace-Beltrami equations, arXiv:1503.04578 (2015).
• V. Fabrikant, Mixed boundary value problems of potential theory and their applications in engineering, Kluwer Academic Pubishers 68, 1991.
• G. Fichera, Analisi esistenziale per le soluzioni dei problemi al contorno misti, relativi all'equazione e ai sistemi di equazioni del secondo ordine di tipo ellittico, autoaggiunti, Annal. Scuola Norm. Super. 1 (1949), 75–100.
• ––––, Sul problema della derivata obliqua e sul problema misto per l'equazione di Laplace Boll. Union. Matem. 7 (1952), 367–377.
• A.S. Fokas, Complex variables: Introduction and applications, Cambridge University Press, Cambridge, 2003.
• G.B. Folland, Introduction to partial differential equations, 2nd edition, Princeton University Press, Princeton, 1995.
• G. Giraud, Problèmes mixtes et Problèmes sur des variétés closes, relativement aux équations linéaires du type elliptique, Imprimerie de l'Université, 1934.
• J. Helsing, Integral equation methods for elliptic problems with boundary conditions of mixed type, J. Comp. Phys. 228 (2009), 8892–8907.
• ––––, Solving integral equations on piecewise smooth boundaries using the RCIP method: A tutorial, in Abstract and applied analysis 2013, Hindawi Publishing Corporation, 2013, http://www.maths.lth.se/na/staff/helsing/Tutor/index.html.
• A. Lorenzi, A mixed boundary value problem for the Laplace equation in an angle, Part 1, Rend. Sem. Matem. Univ. Padova 54 (1975), 147–183.
• E. Magenes, Sui problemi di derivata obliqua regolare per le equazioni lineari del secondo ordine di tipo ellittico, Annal. Mat. Pura Appl. 40 (1955), 143–160.
• W.C.H. McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000.
• C. Pérez-Arancibia and O.P. Bruno, High-order integral equation methods for problems of scattering by bumps and cavities on half-planes, JOSA A 31 (2014), 1738–1746.
• A. Signorini, Sopra un problema al contorno nella teoria delle funzioni di variabile complessa, Annal. Mat. Pura Appl. 25 (1916), 253–273.
• S. Warschawski, On differentiability at the boundary in conformal mapping, Proc. Amer. Math. Soc. 12 (1961), 614–620.
• S.E. Warschawski, On the higher derivatives at the boundary in conformal mapping, Trans. Amer. Math. Soc. 38 (1935), 310–340.
• W.L. Wendland, E. Stephan, G.C. Hsiao and E. Meister, On the integral equation method for the plane mixed boundary value problem of the Laplacian, Math. Meth. Appl. Sci. 1 (1979), 265–321.
• N.M. Wigley, Asymptotic expansions at a corner of solutions of mixed boundary value problems, J. Math. Mech. 13 (1964), 549–576.
• ––––, Mixed boundary value problems in plane domains with corners, Math. Z. 115 (1970), 33–52.
• Y. Yan and I.H. Sloan, et al., On integral equations of the first kind with logarithmic kernels, University of NSW, 1988.
• S. Zaremba, Sur un problème mixte relatif a l'équation de Laplace, Bull. Acad. Sci. (1910), 313–344.