Adaptive finite element solution of coupled PDE-ODE systems
(English)Manuscript (preprint) (Other academic)
We consider an implicit / explicit method for solving a semilinear parabolic partial differential equation (PDE) coupled to a set of nonlinear ordinary differential equations (ODEs). More specifically the PDE of interest is the heat equation where the right hand side couple with the ODEs. For this system, a posteriori error estimates are derived using the method of dual-weighted residuals giving indicators useful for constructing adaptive algorithms.
We distinguish the errors in time and space for the PDE and the ODEs separately and include errors due to transferring the solutions between the equations. In addition, since the ODEs in many applications are defined on a much smaller spatial scale than what can be resolved by the finite element discretization for the PDE, the error terms include possible projection errors arising when transferring the global PDE solution onto the local ODEs. Recovery errors due to passing the local ODE solutions to the PDE are also included in this analysis.
The method is illustrated on a realistic problem consisting of a semilinear PDE and a set of nonlinear ODEs modeling the electrical activity in the heart. The method is computationally expensive, why an adaptive algorithm using blocks is used.
Research subject Mathematics
IdentifiersURN: urn:nbn:se:umu:diva-33866OAI: oai:DiVA.org:umu-33866DiVA: diva2:318483