## UC Berkeley / Lawrence Berkeley Laboratory

#### Blockwise Adaptive Finite Element Solution of a Coupled PDE-ODE System

**August Johansson, UC Berkeley**

##### February 16th, 2011 at 4PM–5PM in 939 Evans Hall [Map]

Systems with a partial differential equation (PDE) coupled to ordinary
differential equations (ODEs) often appear in applications where the
ODEs model microscopic chemical reactions, and the PDE models a
macroscopic behaviour. Such applications include simulation of
combustion, pollution and semiconductors. In this talk, we consider a
PDE–ODE system for the electrical activity of the heart consisting of
a semilinear parabolic PDE coupled to a set of nonlinear ODEs.
Motivated by the fact that the dynamics of the ODEs may be much faster
than that of the PDE, we present a multirate finite element method for
the discretization. To control the error, a posteriori error estimates
are derived using the method of dual-weighted residuals, giving error
indicators useful for constructing adaptive algorithms. The indicators
distinguish the errors in time and space for the PDE and the ODEs
separately and include errors due to the transfer of data between the
PDE and the ODEs. The method is computationally expensive, why a novel
adaptive algorithm using space-time blocks is used.