TITLE: Convex Splitting Schemes for Conserved Bistable Gradient
Equations: Applications in Grain Boundary Motion, Solidification,
Tumor Growth, and Two-Phase Flow

ABSTRACT: Bistable gradient equations (BGEs) are a class of PDEs that
model important biological and physical problems. Loosely speaking,
these equations result as gradient flows with respect to certain
bistable (or biconvex) energies.  Perhaps the best known BGE is the
celebrated Cahn-Hilliard equation.  In the talk I focus on two
important conserved BGEs, the phase field crystal (PFC) equation and
the Cahn-Hilliard-Hele-Shaw (CHHS) equation, which are nonlinear
6th-order and 4th-order PDEs, respectively.  These equations model a
wide range of phenomena, including crystal growth, grain coarsening,
phase-separation, cell motion, two-phase flow, and tumor growth.
Numerical solution of BGEs can pose enormous challenges, because the
equations (i) are typically of high order, (ii) are potentially highly
nonlinear, and (iii) usually must be solved over large space and time
scales.

In the talk I describe convex-splitting schemes for BGEs. The schemes
are simple, general, and powerful, and they possess two important
properties, unconditional unique solvability and unconditional
stability.  The energy stability can often be exploited to prove
various norm stabilities, as well as convergence.  In the talk I will
give details of the convergence analyses for the PFC equation.  The
unique solvability follows from the fact that the schemes are derived
as the gradients of strictly convex functionals.  As a result,
practical (though not necessarily highly efficient) solvers can always
be crafted, since descent methods will converge unconditionally.  The
biggest challenge of this work is in designing truly efficient solvers
for the potentially highly nonlinear CS schemes.  We have had some
early, important successes in this direction, having crafted nearly
optimally efficient nonlinear multigrid solvers for PFC and CHHS
equations, which I will report on.