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.

/pdf/robust-numerical-methods-for-singularly-perturbed-35aj798blf.pdf

Robust Numerical Methods for Singularly Perturbed Differential Equations: A Survey Covering 2008–2012

The second edition of the book [109] appeared many years ago and was for many years a reliable guide into the world of numerical methods for singularly perturbed problems. Since then many new results came into the game, we present some selected ones and the related sources.

Robust Numerical Methods for Singularly Perturbed Differential Equations-Supplements

A semilinear second-order parabolic equation is considered in a regular and a singularly perturbed regime. For this equation, we give computable a posteriori error estimates in the maximum norm. Semidiscrete and fully discrete versions of the backward Euler, Crank--Nicolson, and discontinuous Galerkin $dG(r)$ methods are addressed. For their full discretizations, we employ elliptic reconstructions that are, respectively, piecewise-constant, piecewise-linear, and piecewise-quadratic for $r=1$ in time. We also use certain bounds for the Green's function of the parabolic operator.

/pdf/maximum-norm-a-posteriori-error-estimation-for-parabolic-27a5a7kaqh.pdf

Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions

Linear and semilinear second-order parabolic equations are considered. For these equations, we give a posteriori error estimates in the maximum norm that improve upon recent results in the literature. In particular it is shown that logarithmic dependence on the time step size can be eliminated. Semidiscrete and fully discrete versions of the backward Euler and of the Crank-Nicolson methods are considered. For their full discretizations, we use elliptic reconstructions that are, respectively, piecewise-constant and piecewise-linear in time. Certain bounds for the Green’s function of the parabolic operator are also employed.

Improved maximum-norm a posteriori error estimates for linear and semilinear parabolic equations

In this article, we present a quadratic B-spline collocation method on an exponentially graded mesh for the solution of two-parameter singularly perturbed boundary value problems (TP-SPBVPs) whose ...

Quadratic B-spline collocation method for two-parameter singularly perturbed problem on exponentially graded mesh

— A semilinear second-order singularly perturbed parabolic equation in one space dimension is considered. For this equation, we give computable a posteriori error estimates in the maximum norm for a difference scheme that uses Backward-Euler in time and central differencing in space. Sharp L1-norm bounds for the Green’s function of the parabolic operator and its derivatives are derived that form the basis of the a posteriori error analysis. Numerical results are presented. 2010 Mathematical subject classification: 65M15; 65M50.

/pdf/maximum-norm-a-posteriori-error-estimation-for-a-time-2vwb3oswyc.pdf

Maximum Norm a Posteriori Error Estimation For a Time-dependent Reaction-diffusion Problem

Collocation with biquadratic $$C^1$$
 -splines for a singularly perturbed reaction-diffusion problem in two dimensions is studied. A second-order supremum norm a posteriori error bound is derived for the collocation method on an arbitrary mesh. Numerical results are presented that support our theoretical estimate.

A posteriori maximum-norm error bounds for the biquadratic spline collocation method applied to reaction-diffusion problems

Maximum‐norm a posteriori error bounds for a collocation method applied to a singularly perturbed reaction–diffusion problem in three dimensions

Collocation with quadratic C
 1-splines for a singularly perturbed reaction-diffusion problem in one dimension is studied. A modified Shishkin mesh is used to resolve the layers. The resulting method is shown to be almost second order accurate in the maximum norm, uniformly in the perturbation parameter. Furthermore, a posteriori error bounds are derived for the collocation method on arbitrary meshes. These bounds are used to drive an adaptive mesh moving algorithm. Numerical results are presented.

Approximation of singularly perturbed reaction-diffusion problems by quadratic C 1 -splines

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