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A parallel unified transform solver based on domain decomposition for solving linear elliptic PDEs

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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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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]

$$\begin{aligned} \begin{bmatrix} K_{1}&&E_{1}\\&K_{2}&&E_{2}\\&\ddots&\vdots \\&&K_{s}&E_{s} \\ E_{1}^{T}&E_{2}^{T}&\cdots&E_{s}^{T}&C \end{bmatrix} \begin{bmatrix} u_{1}\\ u_{2}\\ \vdots \\ u_{s}\\ u_{\varGamma } \end{bmatrix}= \begin{bmatrix} f_{1}\\ f_{2}\\ \vdots \\ f_{s}\\ f_{\varGamma }. \end{bmatrix} \end{aligned}$$
(33)

where matrix A can also be written as

$$\begin{aligned} A=\begin{bmatrix} K&E \\ E^{T}&C \end{bmatrix}. \end{aligned}$$
(34)

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]

$$\begin{aligned} (C-E^{T}K^{-1}E)u_{\varGamma }=f_{\varGamma }-E^{T}K^{-1}f, \end{aligned}$$
(35)

where

$$\begin{aligned} S=C-E^{T}K^{-1}E, \end{aligned}$$
(36)

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]:

$$\begin{aligned}&{\text{compute}}\,x=Eu_{\varGamma }\\&{\text{solve}}|\,Ky=x\\&{\text{compute}}\, Cu_{\varGamma }-E^{T}y. \end{aligned}$$
(37)
figure j

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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