## Abstract

A hybrid approach for the solution of linear elliptic PDEs, based on the unified transform method in conjunction with domain decomposition techniques, is introduced. Given a well-posed boundary value problem, the proposed methodology relies on the derivation of an approximate global relation, which is an equation that couples the finite Fourier transforms of all the boundary values. The computational domain is hierarchically decomposed into several nonoverlapping subdomains; for each of those subdomains, a unique approximate global relation is derived. Then, by introducing a modified Dirichlet-to-Neumann iterative algorithm, it is possible to compute the solution and its normal derivative at the resulting interfaces. By considering several hierarchical levels, higher spatial resolution can be achieved. There are three main advantages associated with the proposed approach. First, since the unified transform is a boundary-based technique, the interior of each subdomain does not need to be discretized; thus, no mesh generation is required. Additionally, the Dirichlet and Neumann values can be computed on the interfaces with high accuracy, using a collocation technique in the complex Fourier plane. Finally, the interface values at each hierarchical level can be computed in parallel by considering a quadtree decomposition in conjunction with the iterative Dirichlet-to-Neumann algorithm. The proposed methodology is analysed both regarding implementation details and computational complexity. Moreover, numerical results are presented, assessing the performance of the solver.

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## References

Arabnia HR, Taha TR (1998) A parallel numerical algorithm on a reconfigurable multi-ring network. J Telecommun Syst 10:185–203

Ashton ACL (2013) The spectral Dirichlet–Neumann map for Laplace’s equation in a convex polygon. SIAM J Math Anal 45(6):3575–3591

Babuska I, Guo B (2000) Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two dimensions. Numer Math 85(2):219–255

Balasubramanian P, Arabnia HR (2014) Computation of error resiliency of Muller C-element. In: Proceedings on International Conference on Computational Science and Computational Intelligence, pp 179–180

Bhandarkar SM, Arabnia HR (1995) The REFINE multiprocessor-theoretical properties and algorithms. Parallel Comput 21(11):1783–1806

Bjorck A (2015) Numerical methods in matrix computations. Texts in Applied Mathematics. Springer, Berlin

Canuto C, Hussaini MY, Quarteroni A, Zang TA (2006) Spectral methods. Springer, Berlin

Chan TF, Goovaerts D (1989) Schur complement domain decomposition algorithms for spectral methods. Appl Numer Math 6:53–64

Chazelle B, Dobkin D (1985) Optimal convex decompositions. In: Toussaint G (ed) Computational geometry. North-Holland, Amsterdam, pp 63–133

Colbrook MJ (2018) Extending the unified transform: curvilinear polygons and variable coefficient PDEs. IMA J Numer Anal. https://doi.org/10.1093/imanum/dry085

Colbrook MJ, Flyer N, Fornberg B (2018) On the Fokas method for the solution of elliptic problems in both convex and non-convex polygonal domains. J Comput Phys 374:996–1016

Courant R, Hilbert D (1989) Methods of mathematical physics, vol 1. Wiley, Hoboken

Davis C-IR, Fornberg B (2014) A spectrally accurate numerical implementation of the Fokas transform method for Helmholtz-type PDEs. Complex Var Elliptic Equ 59(4):564–577

Elliotis M, Georgiou G, Xenophontos C (2005) Solving Laplacian problems with boundary singularities: a comparison of a singular boundary integral method with the p/hp version of the finite element method. Appl Math Comput 169:485–499

Fernandez A, Baleanu D, Fokas AS (2018) Solving PDEs of fractional order using the unified transform method. Appl Math Comput 339:738–749

Fokas AS (1997) A unified transform method for solving linear and certain nonlinear PDEs. Proc R Soc Lond Ser A 453:1411–1443

Fokas AS (2001) Two-dimensional linear PDEs in a convex polygon. Proc R Soc Lond Ser A 457:371–393

Fokas AS (2002) A new transform method for evolution PDEs. IMA J Appl Math 67:559–590

Fokas AS (2008) A unified approach to boundary value problems. SIAM, Philadelphia

Fornberg B, Flyer N (2011) A numerical implementation of Fokas boundary integral approach: Laplace’s equation on a polygonal domain. Proc R Soc A 467:2083–3003

Franceschini A, Paludetto Magri V, Ferronato M, Janna C (2018) A robust multilevel approximate inverse preconditioner for symmetric positive definite matrices. SIAM J Matrix Anal Appl 39:123–147

Fulton S, Fokas AS, Xenophontos C (2004) An analytical method for linear elliptic PDEs and its numerical implementation. J Comput Appl Math 167:465–483

Grylonakis E-NG, Filelis-Papadopoulos CK, Gravvanis GA (2015) A note on solving the generalized Dirichlet to Neumann map on irregular polygons using Generic Factored Approximate Sparse Inverses. CMES Comput Model Eng Sci 109(6):505–517

Grylonakis E-NG, Filelis-Papadopoulos CK, Gravvanis GA (2017) A hybrid method for solving inhomogeneous elliptic PDEs based on Fokas method. Comput Methods Appl Math. https://doi.org/10.1515/cmam-2017-0053

Grylonakis E-NG, Filelis-Papadopoulos CK, Gravvanis GA (2018) A class of unified transform techniques for solving linear elliptic PDEs in convex polygons. Appl Numer Math 129:159–180

Grylonakis E-NG, Filelis-Papadopoulos CK, Gravvanis GA, Fokas AS (2019) An iterative spatial-stepping numerical method for linear elliptic PDEs using the Unified Transform. J Comput Appl Math 352:194–209

Grylonakis E-NG, Filelis-Papadopoulos CK, Gravvanis GA, Fokas AS (2019) An adaptive complex collocation method for solving linear elliptic PDEs in regular convex polygons based on the unified transform. Numer Math Theory Methods Appl 12(2):348–369

Hashemzadeh P, Fokas AS, Smitheman SA (2015) A numerical technique for linear elliptic partial differential equations in polygonal domains. Proc Math Phys Eng Sci 471:20140747. https://doi.org/10.1098/rspa.2014.0747

Janna C, Castelletto N, Ferronato M (2015) The effect of graph partitioning techniques on parallel Block FSAI preconditioning: a computational study. Numer Algorithms 68(4):813–836

Jayashree HV, Thapliyal H, Arabnia HR, Agrawal VK (2016) Ancilla-input and Garbage-output Optimized Design of a Reversible Quantum Integer Multiplier. J Supercomput 72(4):1477–1493

Jiri K, Rozloznik M, Tuma M (2017) An adaptive multilevel factorized sparse approximate inverse preconditioning. Adv Eng Softw 113:19–24

Kyziropoulos PE, Filelis-Papadopoulos CK, Gravvanis GA (2018) A class of symmetric factored approximate inverses and hybrid two-level solver. Int J Comput Methods 15(2):1850050

Makaratzis AT, Filelis-Papadopoulos CK, Gravvanis GA (2016) Parallel multilevel recursive approximate inverse techniques for solving general sparse linear systems. J Supercomput 72(6):2259–2282

Mathew T (2008) Domain decomposition methods for the numerical solution of partial differential equations. Springer, Berlin

Moutafis BE, Filelis-Papadopoulos CK, Gravvanis GA (2017) Parallel multi-projection preconditioned methods based on semi-aggregation techniques. J Comput Sci 22:45–54

Moutafis BE, Filelis-Papadopoulos CK, Gravvanis GA (2018) Parallel Schur complement techniques based on multiprojection methods. SIAM J Sci Comput 40(4):634–654

Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes. The art of scientific computing, 3rd edn. Cambridge University Press, Cambridge

Quarteroni A (2014) Numerical models for differential problems (

*MS*&*A*), 2nd edn. Springer, BerlinSaad Y (2003) Iterative methods for sparse linear systems, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia

Sauter SA, Schwab C (2010) Boundary element methods. Springer, Berlin

Sifalakis AG, Fokas AS, Fulton S, Saridakis YG (2008) The generalized Dirichlet–Neumann map for linear elliptic PDEs and its numerical implementation. J Comput Appl Math 219(1):9–34

Toselli A, Widlund O (2005) Domain decomposition methods—algorithms and theory. Springer, Berlin

Valafar H, Arabnia HR, Williams G (2004) Distributed global optimization and its development on the multiring network. Int J Neural Parallel Sci Comput 12(4):465–490

Wang H, Xiang S (2012) On the convergence rates of Legendre approximation. Math Comput 81:861–877

Xi Y, Li R, Saad Y (2016) An algebraic multilevel preconditioner with low-rank corrections for sparse symmetric matrices. SIAM J Matrix Anal Appl 37(1):235–259

Zhu JZ, Zienkiewicz OC (1988) Adaptive techniques in the finite element method. Commun Appl Numer Methods 4:197–204

Zhu Y, Sameh AH (2017) PSPIKE+: a family of parallel hybrid sparse linear system solvers. J Comput Appl Math 311:682–703

Zienkiewicz OC, Taylor OL, Zhu JZ (2013) The finite element method: its basis and fundamentals, 7th edn. Butterworth-Heinemann, Oxford

## Acknowledgements

The authors acknowledge the Greek Research and Technology Network (GRNET) for the provision of the National HPC facility ARIS under project PR004033—ScaleSciCompII, and support from EPSRC, UK. The authors are also thankful to Matt Colbrook for useful suggestions.

## Funding

Funding was provided by Engineering and Physical Sciences Research Council for A.S. Fokas.

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## Appendix: Finite Element Domain Decomposition (DDFEM) algorithm

### Appendix: Finite Element Domain Decomposition (DDFEM) algorithm

Let us consider a linear system Au = f, arising from a finite element discretization of a linear elliptic PDE. For a general decomposition into s subdomains, the linear system has the following structure [39]

where matrix A can also be written as

The vectors \(\left\{ u_{j} \right\} _{1}^{s}\) represent the solution at the interior of the s subdomains, and \(u_{\varGamma }\) represents the solution at the interfaces. Using block Gaussian elimination, the interface values are obtained by solving the following reduced system [39]

where

is called the Schur complement matrix [39]. The reduced system can be solved without explicitly assembling the Schur complement matrix S by considering a Krylov subspace iterative method. The matrix-by-vector operations \(Su_{\varGamma }\) are performed as follows [39]:

In Algorithm 10 the Schur complement, finite element procedure is described. It should be noted that R represents a restriction matrix. Further implementation details can be found in [34].

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Grylonakis, E.N.G., Gravvanis, G.A., Filelis-Papadopoulos, C.K. *et al.* A parallel unified transform solver based on domain decomposition for solving linear elliptic PDEs.
*J Supercomput* **75**, 4947–4985 (2019). https://doi.org/10.1007/s11227-019-02772-2

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DOI: https://doi.org/10.1007/s11227-019-02772-2