## Research Abstracts Online

January - December 2011

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### University of Minnesota Twin Cities

College of Science and Engineering

School of Mathematics

# PI: Douglas N. Arnold, Fellow

### Finite Element Exterior Calculus With FEniCS

These researchers are using the FEniCS finite element software system to study several issues related to the performance of algorithms developed through the finite element exterior calculus (FEEC). The Principal Investigator is the originator of FEEC, and FEniCS, which is one of the most impressive open-source projects in numerical partial differential equations, has been designed so that it is well suited to implementing and testing FEEC-based algorithms. FEniCS, which allows one to implement finite elements from a high level mathematical perspective, includes a Python/C++ problem solving environment which provides access to advanced solver systems like PETSc, uBLAS, and Trilinos, which in turn use MPI, and so are in a position to make efficient use of Itasca. This research group is studying: the performance of a newly developed mixed finite elements with weak symmetry for problems of elastodynamics and dynamical viscoelasticity; the computation of bases of harmonic forms on three-dimensional domains with complex topology; the development of *a posteriori* error estimators and adaptive mesh strategies within FEEC; and the convergence of smoother-than-expected solutions of finite element methods for elliptic problems on domains with reentrant corners; and similar issues. Since starting work at MSI on the Itasca cluster, the researchers have implemented FEniCS and tuned its performance, and are getting very good results for a number of problems.

### Group Members

Po Hu, Graduate Student

John Lee, Graduate Student

Lizao Li, Graduate Student

Bo Yang, Graduate Student