Below is given a summary of the work undertaken in the second year of my PhD

• Talks
• Numerical Methods
• Convex-Concave Splitting
• Hessian based Approach
• Numerical Issues
• Trust Region Method
• PAC Meeting

I have given several talks in my 2nd year more information can be found here:

• 18th March 2014: MasDoc-CCA Conference Talk: Summary of 2nd year work considering the numerical methods used in my PhD, approximately 30 minutes
• 19th May 2014: MasDoc-CCA Conference Talk: Summary of 2nd year work considering the issues with the numerical methods used in my PhD, approximately 30 minutes

We consider various ways of minimising the PFC functional

\( F[u]=∫_\Omega \frac{1}{2} u(Δ+1)^2 u−\frac{\delta}{2}u^2+u^4dx .\)

whilst conserving the average

\( \bar u = \int_\Omega u \textrm{d} x \)

Our methods are based on gradient flow of the functional in a given Norm.

We wish to consider gradient flow in both the \(L^2\) and \(H^{-1}\)norm. In this case we split the energy into a convex and concave part.

\( \mathcal F [u] = \mathcal F_C[u] - \mathcal F_{E} [u].\)

in (S.M. Wise, C. Wang, J.S. Lowengrub An Energy-Stable and Convergent Finite Difference Scheme for the Phase Field Crys- tal Equations, SIAM Journal of Numerical Analysis, Volume 47, Issue 3, pages 2269-2288, (2009).) they use the splitting

\( \mathcal F_C[u] =\frac{1}{2}\| \Delta u \|_{L^2(\Omega)}^2+ \frac{(1-\delta)}{2} \|u \|_{L^2(\Omega)}^2 + \frac{1}{4} \|u \|_{L^4(\Omega)}^4,~~~ \mathcal F_E [u] =\|\nabla u \|_{L^2(\Omega)}^2 \)

However this is very slow to converge as the non-linear term is evaluated implicitly. In (M. Elsey, B. Wirth, A Simple and Efficient Schemer for Phase Field Crystal Simulation, ESIAM: Mathematical Modelling and Numerical Analysis, Volume 47, Issue 5 , pages 1413-1432, (2013).) they develop a scheme in which the non-linear term is treated explicitly

\( \mathcal F_c[u] =\frac{1}{2}\| \Delta u+u \|_{L^2(\Omega)}^2+ \frac{(C-\delta)}{2} \|u \|_{L^2(\Omega)}^2 ,~~~ \mathcal F_e [u] =\frac{C}{2} \|u \|_{L^2(\Omega)}^2 -\frac{1}{4} \|u \|_{L^4(\Omega)}^4\).

which can be shown to converge when the constant \(C\) is sufficiently large.

As an alternative to concave-convex splitting we consider an approach based on a gradient flow. Specifically we take a gradient flow based on Newton's method i.e.

\( \langle \delta^2 \mathcal F [u]v,v \rangle =- \langle \delta \mathcal F [u],v \rangle \)

However in this case the Hessian is not guaranteed to be positive definite so we need to modiy the metric to bbe positive definite. Once this is done we are able to prove stability and convergence.

We wish to consider simulations of realistic materials. In this case a large number of crystals will be involved, this means that we will need a small number of points per crystal in our numerics to make the problem computationally tractable. We experience difficulties in this case.

A possible method to combat the numerical issues mentioned above in the trust region method. In this case the functional is updated by minimising the quadratic function

\( m(s)= \langle \delta \mathcal F [u] ,s \rangle + \langle \delta^2 \mathcal F[u]s,s \rangle \)

subject to the update lying with a region, which is updated given how good the reduction is,

\( \|s\|_A \le \Delta \)

and \( A\) defines a norm. However we need to consider how to constrain the energy so that the average of the variable is conserved, standard Trust region methods in an arbitary norm are found in (T. Steihaug The Conjugate Gradient Method and Trust Regions in Large Scale Optimisation, SIAM Journal of Numerical Analysis, 20:3, pages 626-637, (1983).) and for constrained norms are (R.H. Byrd,R.B, Schnabel, G.A. Shultz. A Trust Region Algorithm for Nonlinearly Constrained Optimisation, SIAM Journal of Numerical Analysis, 24:5, pages 1152-1169, (1987)).

During the third term progress is assessed via a PAC meeting.