We mainly present the construction of meshes suitable for a one-dimensional second order convection-diffusion problem posed in with a layer term (of course, the layer could also be located at
or at both endpoints of the interval, for instance, in case of a reaction-diffusion problem). For 2D problems with boundary layers, tensor product ideas allow to extend the 1D principles of the mesh construction to the two-dimensional case.
For finite difference methods it seems natural to reach for uniform convergence in the discrete maximum norm; that is, the computed solution satisfies
for some positive constants and that are independent of and . A power of is a suitable measure of the error for the particular families of meshes that we will discuss, but a bound of this type is inappropriate for an arbitrary family of meshes; see .
The aim to achieve uniform convergence in the maximum norm is demanding and leads even in 1D to meshes which sometimes do not have desirable properties. Therefore, we will take into account as well meshes where the constant in (1.1) will weakly depend on . Such a desirable property is, for example, the local quasi-uniformity of the mesh.
For finite element methods in 2D it is unrealistic to hope for uniform convergence in the maximum norm on very non-regular meshes. Therefore, one analyzes finite element methods in scaled Sobolev space norms. To achieve the
-independence of the constants in the estimates again layer-adapted meshes are necessary.
2 Bakhvalov 1969
Bakhvalov’s original mesh  uses mesh points near defined by
in some interval , here is a parameter. More precisely, Bakhvalov proposed the mesh-generating function
Remarkably, is defined by the requirement that the mesh-generating function is . Thus, has to solve the nonlinear equation
On that mesh, Bakhvalov studied the finite difference method for reaction-diffusion problems and proved uniform convergence of the second order.
Surprisingly, more than 10 years nobody cited Bakhvalov’s paper. In Russian, Vassiljeva was the first to refer to Bakhvalov in 1982, in English written papers Blatov, Boglaev and Liseikin quoted Bakhvalov 1990 ( papers of Vulanovic since 1983 based on Bakhvalov’s work appeared in a small unknown journal in Novi Sad, Serbia).
Boglaev, Liseikin and Vulanovic presented different versions of Bakhvalov’s mesh. For instance, a Bakhvalov-type mesh is given by
In the mesh is equidistant, where the transition point from the fine to the coarse mesh is defined by
Bakhvalov-type meshes are simpler than the Bakhvalov meshes and the mesh-generating function is not longer . But both meshes are not locally quasi-equidistant. In some cases for these meshes and finite difference methods in 1D optimal error estimates are known, but the analysis is often more complicated than for Shishkin-type meshes (see Section 5). For linear finite elements in 1D an optimal error estimate was proved 2006 in , but is open in 2D (see ).
As Linß pointed out in , a Bakhvalov-mesh can also be generated by equidistributing the monitor function
In several papers (see  and its references) Liseikin examines the convergence of finite difference methods when using mesh generating functions of the given independent variable that satisfy for all . This approach generates a graded grid of Bakhvalov type. His book  develops a general theory of grid generation. The analysis in these sources is written in terms of “layer-resolving transformations”; their relationship to mesh generating functions in a singular perturbation context is discussed in .
In  we find the proposal to generate a mesh by the implicitly defined function
The mesh has the advantage that it is not necessary to use different mesh generating functions in different regions. But (2.3) is not so easy to solve, however, a solution based on the use of Lambert’s W-function is possible. Some difference schemes (and finite elements) on that mesh can be analyzed similarly as on Bakhvalov-type meshes .
3 van Veldhuizen 1978
Van Veldhuizen  studied a second order 1D convection-diffusion problem with constant convection term and proved for the finite element method with -th order polynomials and Radau quadrature for
Here the interval is decomposed into and , and and are the maximal step sizes in the corresponding intervals (for he proved a first order result).
This estimate motivated Veldhuizen to choose
That means, 10 years before Shishkin Veldhuizen proposed to choose ,,Shishkin’s” transition point and provided us with the tools to analyse the finite element method on a Shishkin mesh (see Section 4) !
Although published in the journal ,,Numerische Mathematik”, almost nobody observed the important paper of Veldhuizen, and up to today we have only 5 citations. Moreover, in his numerical experiments Veldhuizen used a Bakhvalov-Shishkin mesh (we will discuss these meshes in Section 6), without knowing Bakhvalov’s paper from 1969.
4 Gartland 1988
Gartland  studied higher order finite difference methods in 1D and graded a mesh in the following way:
The restriction ensures that the mesh is locally quasi-equidistant.
If simple upwinding for a convection-diffusion problem is uniformly convergent in the sense of (1.1) for some constant , and the mesh is locally quasi-equidistant (uniformly in ), then the number of mesh intervals must increase as . To see this, observe that the arguments of  are still valid when slightly modified by considering a limit as with and ; one then arrives at the conclusion of that paper that . (There are some minor extra mesh assumptions such as existence of and .) But the mesh diameter is at least , so the locally quasi-equidistant property implies that , where is the constant in
Hence , so .
Introducing the transition points by
Gartland observed that the number of mesh points in the inner region as well in the outer region is of order , but in the transition region of order . On that mesh Gartland proved the uniform convergence of some finite difference schemes.
We call the modification of the mesh where (4.1) is replaced by
Gartland-type mesh. The number of mesh points is now independent of and the mesh allows optimal error estimates. The mesh is not locally quasi-eqidistant.
5 Shishkin 1988
Shishkin spread the great idea to use the very simple piecewise constant meshes in combination with the transition point from the fine to the coarse mesh defined by
Consequently, for small , one has , and is chosen in dependence of the order of the method used.
Then, each of the intervals and is subdivided equidistantly in subintervals. It is not vital that one has exactly the same number of subintervals in and . All that the theory demands is that as the number of subintervals in each of these two intervals is bounded below by for some constant .
The coarse part of this Shishkin mesh has spacing , so . The fine part has spacing , so . Thus there is a very abrupt change in mesh size as one passes from the coarse part to the fine part. The mesh is not locally quasi-equidistant, uniformly in .
Before Shishkin’s work uniform finite difference methods were often based on the properties of pointwise uniform consistency and uniform stability in the maximum norm. But pointwise uniform consistency is not necessary. While Shishkin used the maximum principle and barrier functions in the analysis, other approaches use improved stability properties (Andreev, Kopteva, Linß 1996-98 [1, 2, 22]).
In 1D, we know uniform second order finite difference schemes on Shishkin meshes, but in 2D mostly only the first order upwind scheme is analyzed (for convection diffusion problems).
Finite element methods on Shishkin meshes in 1D were first studied 1995 by Sun and Stynes , the analysis for second order problems was also published in the two books [32, 38] from 1996. But the more important analysis in 2D was still in open problem, because in 2D the analysis of finite element methods is traditionally based on shape regular or isotropic meshes (see Section 6).
If a method for a problem with a smooth solution has the order , due to the fine mesh size in the case we can expect that the error on a Shishkin mesh is of the order . Especially for higher order methods the logarithmic factor is troublesome. An optimal mesh should generate an error of the order . This is the reason for introducing S-type meshes, see Section 7.
Shishkin meshes and S-type meshes have many advantages, but there are also disadvantages: the robustness (see Section 9) and the loss of the possibility to use some typical ingredients in the analysis of finite element methods on isotropic or local uniform meshes.
6 Apel and Dobrowolski 1992/97: anisotropic meshes
In the classical theory of finite element methods one uses the Lagrange interpolanton some element for and its approximation property
Here is the diameter of and the length of the largest ball inscribed . The classical theory assumes a bounded aspect ratio
which excludes anisotropic elements.
In 1992 Apel and Dobrowolski  proved sharp anisotropic interpolation error estimates in the case , Apel and Lube extended 1994 the results to general . Finally in 1999 Apel presented a general theory of anisotropic elements in his famous book .
We just sketch in a simple situation an anisotropic result: Suppose that each element (triangle or rectangle) of a mesh is contained in a rectangle with side lengths and contains a rectangle with side lengths for some fixed constant . In the case of triangles, assume also a maximum angle condition: the interior angles of every mesh triangle are bounded away from . (Triangular Shishkin meshes have maximum angle and consequently satisfy this condition.) Then there exists a constant such that
where we set and .
1997 Dobrowolski created the idea to use these estimates for the first analysis of a singularly perturbed convection-diffusion problem, using linear or bilinear finite elements on a Shishkin mesh in 2D . Parallel and independently Stynes and O’Riordan used a very special technique for bilinear elements .
Since that time anisotropic interpolation error estimates are a standard ingredient to analyze finite element methods on layer-adapted meshes.
7 S-type meshes 1999
In 1999 Linß introduced Bakhvalov-Shishkin meshes similarly as van Veldhuizen 20 years before and analyzed finite difference and finite element methods. More generally, we defined in  the class of S-type meshes. In the fine subinterval with the transition point from the fine to the coarse mesh we use a mesh-generating function. Assuming the function to be strictly increasing, set
We call such meshes Shishkin-type meshes. It turns out that in error estimates for Shishkin-type meshes often the factor appears, where is the mesh-characterizing function defined by
For the original Shishkin mesh we have . A popular optimal mesh is the Bakhvalov-Shishkin mesh with
The mesh points of the fine mesh are given by
Another optimal mesh is the Vulanovic-Shishkin mesh, for details and other possibilities to choose , see .
For higher order finite elements one gets for optimal meshes with bounded the optimal error estimate in the energy norm
which is for higher much better than the result on a Shishkin mesh. One can also try to optimize or the error constants in the estimates .
Further modifications of Shishkin meshes due to Vulanovic are also described in .
In  from 2017 we find a generalization of Shishkin-type meshes based on the property
with some additional parameter . This allows to characterize the so called eXp-mesh  from Xenophontos 2002 as generalized Shishkin-type mesh.
8 Kopteva and Stynes 2001: Adaptively generated meshes
For a long time monitor functions are a standard tool to generate meshes, for instance, the function
The breakthrough came with the results 2001 of Kopteva on a posteriori error bounds for some numerical methods for convection-diffusion problems in 1D. For a conservative form of the upwind finite difference method Kopteva proved
Here is the linear interpolant of the computed solution.
Setting for , Stynes and Kopteva introduced the equidistribution problem: Find , with the computed from the by means of the upwind scheme, such that
Unfortunately, this is a nonlinear problem. But Stynes and Kopteva presented an algorithm which stops in less than steps, such that
where is the error in the solution computed by the algorithm.
Experimental evidence shows that the final mesh computed by the algorithm is strikingly close to a Bakhvalov mesh inside the boundary layer; see [21, Fig. 2]. In contrast, most adaptive algorithms will not generate a mesh resembling a Shishkin mesh.
Unfortunately, in 2D the situation is very different.
An adaptive procedure designed for problems with layers should include an anisotropic refinement strategy. While several anisotropic mesh adaptation strategies do exist, all are more or less heuristic.We do not know of any strategy for convection-diffusion problems in two dimensions where it is proved that, starting from some standard mesh, the refinement strategy is guaranteed to lead to a mesh that allows robust error estimates.
A necessary tool is a robust error estimator on anisotropic meshes; we shortly sketch the situation. Very important is the robustness of the estimators, even on isotropic meshes. In  Sangalli proves the robustness of a certain estimator for the residual-free bubble method applied to convection-diffusion problems. The analysis uses the norm
Although Sangalli’s approach is devoted to residual-free bubbles, the same analysis works for the Galerkin method and the SDFEM. For the convection-diffusion problem, the residual error estimator is robust with respect to the dual norm; see .
The norm above is defined only implicitly by an infinite-dimensional variational problem and cannot be computed exactly in practice. In  Sangalli pointed out that the norm (8.2) seems to be not optimal in the convection-dominated regime. He proposes an improved estimator that is robust with respect to this natural norm  for the advection-diffusion operator, but studied only the one-dimensional case. The relation to another new improved dual norm is studied in detail in .
Today the dual norm or its modification plays an important role in many papers on robust a posteriori error estimation for convection-diffusion problems.
In , Tobiska and Verfürth proved in the dual norm that the same robust a posteriori error estimator can be used for a range of stabilized methods such as streamline diffusion, local projection schemes, subgrid-scale techniques and continuous interior penalty methods. Nonconforming methods are studied in . Variants of discontinuous Galerkin methods are discussed in [12, 15, 28, 44, 56]. Vohralik  presents a very general concept of a posteriori error estimation based on potential and flux reconstructions.
For reaction-diffusion problems it is unclear that the energy norm is a suitable norm for these problems because for small it is unable to distinguish between the typical layer function of reaction-diffusion problems and zero. It would be desirable to get robust a posteriori error estimates in a stronger norm, for instance, some balanced norm or the norm.
The first result with respect to the maximum norm is the a posteriori error estimate of Kopteva  in 2008 for the standard finite difference method on an arbitrary rectangular mesh. Next we sketch the ideas of  from Demlov and Kopteva for a posteriori error estimation for finite elements of arbitrary order on isotropic meshes in the maximum norm 2016.
Using the Green’s function of the continuous operator with respect to a point , the error in that point can be represented by
For some we obtain
Integration by parts yields
Choosing for the Scott-Zhang interpolant of , one needs sharp estimates for to control the interpolation error. These are collected in Theorem 1 of . Thus, one obtains finally with (the constant is of order and )
On anisotropic meshes, in 2015 Kopteva also derived an a posteriori error estimator in the maximum norm , now for linear finite elements. Suppose that the triangulation satisfies the maximum angle condition. Then the first result of  gives
Here is the patch of the elements surrounding some knot of the triangulation, the diameter of . In a further estimator the second term of (8.4), which has isotropic character, is replaced by a sharper result with more anisotropic nature.
To prove (8.4) two difficulties arise. First, it is necessary to use scaled trace inequalities. Moreover, instead of using the Scott-Zhang interpolant of the Green’s function (whose applicability is restricted on anisotropic meshes) Kopteva uses some standard Lagrange interpolant for some continuous approximation of . But the construction is based on the following additional assumption on the mesh. Let us introduce and with some positive . Then, the additional assumption requires that the distance of and is at least some with .
The last condition excludes an too abrupt change of the mesh size, typically for Shishkin meshes. But other layer-adapted meshes satisfy that condition, for instance, Bakhvalov meshes or Bakhvalov-Shishkin meshes.
Unfortunately, we still miss an adaptive strategy based on these estimators leading to optimal meshes.
9 Duran and Lombardi 2006: simple recursively graded meshes
Very simple is the Duran-Lombardi mesh  defined by
Here M is chosen such that but , assuming that the last interval is not extremely small.
The mesh is locally quasi-equidistant and glitters by its simplicity. Almost uniform error estimates with respect to are possible, but the number of mesh-points is proportional to . The finite element analysis presented in  differs a bit from the analysis on a S-type mesh. It has the advantage to require only estimates for derivatives (in contrast to a solution decomposition with estimates for the components of the decomposition).
For finite difference methods an analysis on a DL mesh is not known. But for the upwind method, using
(see ) one can simple conclude
Practically, the mesh has two important advantages in comparison to S-type meshes. First, there is no need to define a transition point. And, remarkably, the mesh has the following robustness property: a mesh defined for some can also be used for larger values of the parameter. More precisely: If is the perturbation parameter in the equation which varies, we define a mesh based on a smaller value . Then in a certain range the error is smaller than using to define the mesh. Numerical experiments and some heuristic arguments show optimality for .
Of course, one can also define a mesh by a general recursive formula as
and analyze the necessary properties of to obtain nice convergence results. One possibility is to define a mesh-generating function by interpolation of the values given in the mesh points and then to analyze discretization methods similarly as methods on a S-type mesh, see, for instance, .
10 What else is there?
10.1 Emelyanov and Sidorov Grids
Emelyanov used 1995 the ,,optimal” grids developed by Sidorov 1966 to prove the uniform convergence of some difference schemes . Sidorov’s intention was to construct meshes with the property
Instead solving (10.1), he solved introducing the continuous problem
The exact solution of that problem is known. Moreover, Sidorov meshes have the nice property , useful for the analysis of difference schemes.
Emelyanov used a fine subinterval at the layer and a Sidorov mesh there and a coarse subinterval with and to prove uniform convergence. It would be interesting to know whether or not one can choose in a convection-diffusion problem the first and the last mesh size in such a way that the direct application of Sidorov’s approach leads to uniform convergence.
10.2 Admissibly graded meshes by Liu and Xu
Liu and Xu study 2009 the Galerkin method for a one-dimensional 2m-th order convection-diffusion problem with Hermite splines of degree . For simplicity, we sketch their ideas for the mesh construction in the case . The mesh is called admissible, if and
For other values of a further condition is required. The condition (10.3) is sufficient to prove that the linear interpolant of the layer part satisfies
Next, Liu and Xu show that the condition
with is sufficient for the admissibility of the mesh. Condition (10.4) remembers us of a Gartland-type mesh.
10.3 hp meshes
When analyzing hp finite element methods for singularly perturbed problems it is common to use an hp boundary layer mesh, see . For such methods it is possible to prove exponential convergence.
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