Anisotropic meshes: challenging, but very practical

By Natalia Kopteva http://www.staff.ul.ie/natalia/

Being a theoretical numerical analyst, I’d like to think that our theoretical findings may be practical. After all, according to Ludwig Boltzmann, “nothing is more practical than a good theory”, so there is hope. Being a theoretical numerical analyst within the MACSI group, I manage to occasionally help my colleagues with their computations, and this brings fun and satisfaction. But frequently applications of what I’m working on are not around the corner just yet. For example, in recent years I’ve spent some time working on the so-called a posteriori error estimation on anisotropic meshes. What does this mean?

When one wants to solve a differential equation numerically, the domain is typically partitioned into very small subdomains, such as triangles or quadrilaterals in 2d.  Such partitions are referred to as meshes, or grids. Anisotropic meshes are those that employ extremely narrow mesh elements (for example, extremely narrow triangles). When you expect your solution to change very rapidly in a narrow region (i.e. it exhibits a sharp boundary or interior layer), anisotropic meshes are a no-brainer, as they yield a significant economy of computer memory and time (see the figures below).

At the same time, we may not know where such narrow regions occur, so one wants the numerical code to detect such regions automatically and create suitable anisotropic meshes in such regions. How?? Well, you may start with a computed solution on an unsophisticated mesh, and looking at, for example, its gradients (note that there are more sophisticated metrics!) construct a better mesh. Various ad hoc approaches have been employed, but if one wants a truly reliable algorithm, sharp theoretical bounds are needed for the error of the computed solution in terms of values obtained in the computation process. Such estimates are called a posteriori error estimates; their main purpose is to provide a theoretical framework for automated construction of suitable computational meshes.

Since the pioneering papers by I. Babuška and W. C. Rheinboldt appeared in 1978, the a posteriori error estimation on isotropic meshes has seen a dramatic development. Furthermore, related adaptive-mesh-construction algorithms have been theoretically shown to converge and hence yield reliable computed solutions.

However, the situation is not as satisfactory when it comes to anisotropic meshes. You may wonder why? Surely if one works hard and very carefully traces all the constants in the analysis, a reasonable sharp error bound may be derived? Unfortunately it is not quite like this for a simple reason that the standard theoretical framework used in the analysis of finite element methods, is typically not very suitable for anisotropic meshes. Not only one needs to work harder, but we also need to bring new theoretical tricks to the game. This makes the analysis of anisotropic meshes a very challenging theoretical problem, but it also makes it very interesting!

Bibliography

1. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite elementcomputations, SIAM J. Numer. Anal., 15 (1978), 736-754.
2. Babuška and W. C. Rheinboldt, A-posteriori error estimates for the finiteelement method, International Journal for Numerical Methods in Engineering, 12(1978), 1597-1615.
3. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, Wiley-Interscience, New York, 2000.
4. Kopteva, Maximum-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes, SIAM J. Numer. Anal., 53 (2015), 2519–2544.
5. Kopteva, Energy-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes, Numer. Math., 137 (2017), 607-642.