Numerical modelling, error estimation and applications

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Mathematical models for spatial phenomena such as those used in fluid dynamics and cardiac electrophysiology lead to partial differential equations (PDE). Usually there is a diversity of models to analyze the same phenomena. It is hard to decide which model can provide valid results, how far is the solution of a simplified model compared to a more complex model, and under what circumstances the modelling error is small. Moreover, the associated PDE can rarely be solved analytically. One must rely on numerical methods for solving the PDE problems, leading to so-called "numerical models". Accurate predictions obtained from numerical models are based on the ability to control the modelling error, for instance by properly setting parameters in the model, and also on efficient numerical methods that can provide solutions with controlled accuracy and manageable computational resources. While there are error estimates available for improving the accuracy of numerical solutions, research is still needed for handling the numerical error for time-dependent and complex nonlinear problems. Similarly, error estimates that can be used to control the modelling error are not as commonly available. The objectives of the proposed research program are to develop realistic, reliable and efficient numerical models in cardiac electrophysiology and bio-fluid dynamics. Realism will come from the selection and development of appropriate models, published biophysical knowledge, patient-based geometries and the inclusion of sub-component models. Reliability will be based on estimators for the error coming from different sources in numerical models and on the effective use of these estimators to improve numerical methods and the accuracy of solutions. These sources of error include the modelling error, the error resulting from discretization in time and space, and from the approximate solution of linear and nonlinear systems. Efficiency will come from the selection and development of numerical methods that accounts for specific features of the PDE models, are adaptive in space and time, and coupled with up-to-date iterative solvers. Aside from training 3 MSc and 8 PhD students, the research program will significantly contribute to the advancement of numerical methods for prevalent advection-diffusion-reaction systems and from a more practical standpoint to the development efficient and accurate numerical models of the heart and airways.