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Sangalli, Giancarlo

Robust a-posteriori estimator for advection-diffusion-reaction problems. (English) Zbl 1130.65083

Math. Comput. 77, No. 261, 41-70 (2008).
The paper deals with an a posteriori estimator for a Galerkin discretization of the Dirichlet boundary value problem \[ {\mathcal L}u:=-\varepsilon u^{\prime \prime}+\beta u^{\prime}+\rho u=f(x), \; x \in (0,L), \quad u(0)=u(L)=0 \] with \(\varepsilon >0\) and \(\beta, \rho \geq 0\). The numerical error is measured with respect to a norm possessing the structure like \[ \| \cdot \| _V \approx (\varepsilon | \cdot | _{H_0^1}^2+| \beta| | \cdot | ^2_{1/2}+\rho \| \cdot\| _{L_2}^2)^{1/2}, \] where \(| \cdot | _{1/2}\) is a seminorm of order 1/2. The a posteriori estimator includes the residual and the first derivative of the approximate solution and is robust up to a factor \(\sqrt{\log{(Pe)}}, \) where \(Pe \) is the global Péctlet number \(Pe=| \beta| L/\varepsilon\). Numerical tests in one dimension confirm the theoretical results and the performance of the estimator. A possible two-dimensional extension of the results under consideration is discussed.
Reviewer: Iwan Gawriljuk (Eisenach)

Cited in 2 Reviews
Cited in 26 Documents

MSC:

65L70 Error bounds for numerical methods for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations

Keywords:

advection-diffusion-reaction problems; a-posteriori robust error estimator; finite-element method; Galerkin method; numerical examples
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\textit{G. Sangalli}, Math. Comput. 77, No. 261, 41--70 (2008; Zbl 1130.65083)
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