Robust Numerical Methods for Singularly Perturbed Differential Equations: A Survey Covering 2008–2012
By Hans-Görg Roos. 2012
440 citations recorded by the source corpus.
We present new results in the numerical analysis of singularly perturbed convection-diffusion-reaction problems that have appeared in the last five years. Mainly discussing layer-adapted meshes, we present also a survey on stabilization methods, adaptive methods, and on systems of singularly perturbed equations.
Subfields: Computational Theory and Mathematics, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/772213addbae939add6c511e599876197fc101ca
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In Computational Theory and Mathematics
In Numerical Analysis
Citations in the corpus
- Shishkin meshes in the numerical solution of singularly perturbed differential equations (2010, neutral): “We refer the reader to other sources [55, 72, 73, 95] for more extensive reviews of layer-adapted meshes.”
- A domain decomposition method for solving singularly perturbed parabolic reaction‐diffusion problems with time delay (2018, background): “[14]), the domain D is decomposed into Dp = Ωp × (0, T ], p = �, m, r, where Ω� = (0, 2σ), Ωm = (σ, 1 − σ), Ωr = (1 − 2σ, 1) with”
- Convergence and stability in balanced norms of finite element methods on Shishkin meshes for reaction‐diffusion problems (2015, neutral): “(9) Under these compatibility conditions the following estimates for the interpolation error of the Lagrange interpolant hold true (see [4] or [17]): ‖u − u ‖0 N−2 , ε1/4 |u − u |1 N−1 ln N (10) and ‖u − u ‖∞,Ω0 N−2 , ‖u − u ‖∞,Ω\Ω0 (N−1 ln N)2 , (11)”
- On the role of the Helmholtz decomposition in mixed methods for incompressible flows and a new variational crime (2014, neutral): “emplyoing finite elements of polynomial order k for the discrete velocities and finite elements of order k − 1 for the discrete pressure [8, 30, 43].”
- A higher order difference method for singularly perturbed parabolic partial differential equations (2018, background): “Properties of continuous solution The continuous solution and its derivatives satisfy the following bounds, which can be shown by extending the analysis given in [5,24]: ∣∣∣ ∂p∂q ∂xp∂tq u(x, t) ∣∣∣ ≤ C(1 + ε−p exp (− αx/ε)), for (x, t) ∈ , and p + 2q ≤ 8, (4)”
- A class of discontinuous Petrov-Galerkin methods. Part III (2012, neutral): “The L(2)-error in u for the classical Bubnov-Galerkin method is bounded uniformly by the best approximation error of u measured in the H(1) norm, but the method is not robust as the best approximation error measured in H(1) norm blows up with (see [25] for a detailed discussion).”
- A projection-based stabilized finite element method for steady-state natural convection problem (2011, neutral): “Various stabilization techniques of finite element methods with more satisfactory performance have been developed in [25].”