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## Atomistic/Continuum Multiscale Methods
My main research interest is coarse graining of atomistic models for solids, in particular the construction and analysis of continuum approximations and of atomistic-to-continuum coupling methods (a/c methods, quasicontinuum methods). The idea of a/c methods is to use computationally expensive atomistic models to describe only those regions of a computational domain that require atomistic accuracy, e.g., the neighbourhood of a crystal defect, and to use a coarse-grained continuum model to describe the elastic fields. At least in principle, this process can yield models with near atomistic accuracy at a significantly reduced computational cost. My focus so far has been on the theoretical foundations of these coarse graining techniques.
Together with my collaborators I have mainly worked on the error analysis of different variants of a/c methods. For example, the above figures show the computation of a micro-crack in a 2D model problem. The error graphs (which are consistent with our theory) show how different a/c methods lead to different convergence rates (including O(1)). This is one of the simplest crystal defects, and much more needs to be done to verify the effectiveness of a/c methods.
Some **open Problems** I am still interested in:
* Error estimates for static problems involve modelling errors (consistency errors, variational crimes),
coarsening errors (approximation results), and **stability**
(coercivity). The last ingredient is still poorly
understood for *sharp-interface* (as opposed to diffuse interface) coupling methods.
* Temperature: it should now be possible to
initiate the development and analysis of finite temperature
a/c hybrid models, by employing the techniques of statistical
mechanics.
* QM/MM, MM/MM, QM/CM hybrid methods: A/c methods are only one
example of molecular scale multiscale methods. Other important
techniques, which still require much work, are methods for
concurrent coupling of quantum mechanics with molecular
mechanics or continuum mechanics, or even the coupling of
different molecular mechanics models of different complexity
and accuracy. Force mixing is a popular "solution" to this
problem, so the main challenge would be to construct "good"
energy-based methods.

## Nonlinear Elasticity and the Lavrentiev Gap Phenomenon
In the past I have worked on the numerical approximation of nonlinear elasticity, understood as an energy minimisation problem,
$$
\bar{u} \in \arg\min\big\{ I[u] \big| u \in A \big\}, \qquad \text{where} \qquad I[u] = \int_\Omega W(x, u , \nabla u)\,dx
$$

and $A$ is some *admissible set*. A major difficulty in this area is the **Lavrentiev gap phenomenon** which is said to occur if
$$
\inf \big\{ I[u] \big| A \cap W^{1,\infty} \big\} \gneq \inf \big\{ I[u] \big| u \in A \big\}.
$$

That is, for certain singular variational problems the infimum of the energy taken over Lipschitz functions can be strictly larger than the infimum taken over the entire admissible class. This means, in particular, that conforming finite element methods are incapable of detecting the global minimizers. I have shown [\[18\]](index.php?page=publications#2011-SINUM_ANCFEM), [\[20\]](index.php?page=publications#2011-IMAJNA-lavrentiev_nc) that at least for convex problems (unfortunately this excludes elasticity) one can overcome this by using the non-conforming Crouzeix-Raviart finite element space.
The following figure shows a problem proposed by Foss, Hrusa and Mizel, solved by (mesh-adaptive versions of) the conforming P1 finite element method and the non-conforming Crouzeix-Raviart finite element methods. The left graph clearly shows a gap between the minimal energy (plotted against number of degrees of freedom in the FEM mesh).
Although I have not worked on this topic for some time, there is a particularly difficult **open problem** which I would love to solve, or see solved:
* Construct a numerical method that is capable of approximating global minimizers for any well-posed variational problem (e.g., fitting within the [Ball, 1977] theory) with polyconvex stored energy function.
## Miscellaneous
I also work, or have worked, on the following topics:
* Adaptive finite element methods for phase field models of fracture
* Phase field models for dynamic crack propagation
* Griffith model for fracture
* Discontinuous Galerkin finite element methods
* Gradient flows, in particular nonlinear viscoelasticity
* Large-scale optimisation
* Convergence of adaptive finite element methods
* Numerical enclosure methods for elliptic [\[5\]](index.php?page=publications#2009-SINUM-ApostEx) and parabolic problems
* Optimal control