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- Adriani, Andrea, et al.
(author)
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Generalized Locally Toeplitz matrix-sequences and approximated PDEs on submanifolds : the flat case
- 2023
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In: Linear and multilinear algebra. - : Informa UK Limited. - 0308-1087 .- 1563-5139. ; , s. 1-23
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Journal article (peer-reviewed)abstract
- In the present paper, we consider a class of elliptic partial differential equations with Dirichlet boundary conditions where the operator is the Laplace-Beltrami operator Δ over Ω¯, Ω being an open and bounded submanifold of Rν, ν=2,3. We will take into consideration the classical Pk Finite Elements, in the case of Friedrichs-Keller triangulations, leading to sequences of matrices of increasing size. We are interested in carrying out a spectral analysis of the resulting matrix-sequences. The tools for our derivations are mainly taken from the Toeplitz technology and from the rather new theory of Generalized Locally Toeplitz (GLT) matrix-sequences. The current contribution is only quite an initial step, where a general programme is provided, with partial answers leading to further open questions: indeed the analysis is performed on special flat submanifolds and hence there is room for wide generalizations, with a final picture which is still unclear with respect to, e.g. the role of the submanifold curvature.
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