Mimetic least-squares spectral/hp finite element method for the Poisson equation. (English) Zbl 1280.65130

Palha, Artur; Gerritsma, Marc

Lirkov, Ivan (ed.) et al., Large-scale scientific computing. 7th international conference, LSSC 2009, Sozopol, Bulgaria, June 4–8, 2009. Revised papers. Berlin: Springer (ISBN 978-3-642-12534-8/pbk). Lecture Notes in Computer Science 5910, 662-670 (2010).

Summary: Mimetic approaches to the solution of partial differential equations (PDEs) produce numerical schemes which are compatible with the structural properties – conservation of certain quantities and symmetries, for example – of the systems being modelled. Least squares (LS) schemes offer many desirable properties, most notably the fact that they lead to symmetric positive definite algebraic systems

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, which represent an advantage in terms of computational efficiency of the scheme. Nevertheless, LS methods are known to lack proper conservation properties which means that a mimetic formulation of LS, which guarantees the conservation properties, is of great importance. In the present work, the LS approach appears in order to minimise the error between the dual variables, implementing weakly the material laws, obtaining an optimal approximation for both variables. The application to a 2D Poisson problem and a comparison will be made with a standard LS finite element scheme.
For the entire collection see [Zbl 1204.65003].

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J15	Second-order elliptic equations