Generalized Locally Toeplitz matrix-sequences and approximated PDEs on submanifolds : the flat case

• 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.

Ämnesord

NATURVETENSKAP  -- Matematik -- Beräkningsmatematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Computational Mathematics (hsv//eng)

Nyckelord

Toeplitz sequences

discretization of PDEs

spectral distribution of matrix-sequences

Beräkningsvetenskap med inriktning mot numerisk analys

Scientific Computing with specialization in Numerical Analysis

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