Analysis of the spectral symbol associated to discretization schemes of linear self-adjoint differential operators. (English) Zbl 07380433
Summary: Given a linear self-adjoint differential operator \(\mathscr{L}\) along with a discretization scheme (like Finite Differences, Finite Elements, Galerkin Isogeometric Analysis, etc.), in many numerical applications it is crucial to understand how good the (relative) approximation of the whole spectrum of the discretized operator \(\mathscr{L}^{(n)}\) is, compared to the spectrum of the continuous operator \(\mathscr{L}\). The theory of Generalized Locally Toeplitz sequences allows to compute the spectral symbol function \(\omega\) associated to the discrete matrix \(\mathscr{L}^{(n)}\). Inspired by a recent work by T. J. R. Hughes and coauthors, we prove that the symbol \(\omega\) can measure, asymptotically, the maximum spectral relative error \(\mathscr{E}\geq 0\). It measures how the scheme is far from a good relative approximation of the whole spectrum of \(\mathscr{L}\), and it suggests a suitable (possibly non-uniform) grid such that, if coupled to an increasing refinement of the order of accuracy of the scheme, guarantees \(\mathscr{E}=0\).
MSC:
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 35P05 | General topics in linear spectral theory for PDEs |
| 65N99 | Numerical methods for partial differential equations, boundary value problems |
| 65D25 | Numerical differentiation |
| 65F99 | Numerical linear algebra |
Keywords:
spectral symbol; discretization schemes; linear self-adjoint differential operators; maximum spectral relative errorReferences:
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