Center for Nonlinear Studies

Monday, August 18, 200810:00 AM - 11:00 AMCNLS Conference Room (TA-3, Bldg 1690)
Seminar
Numerical Analysis Seminar
Adaptive Multi-Mesh hp-FEM for Time-Dependent Multi-Physics Coupled Problems
P. SolinUniversity of Texas at El Paso
Our work is related to the design and analysis of automatic hp-adaptive algorithms for multiphysics coupled problems described by systems of nonlinear and time-dependent PDEs. Interacting physical fields often exhibit large qualitative and quantitative differences, and thus approximating all of them on the same finite element mesh can be highly inefficient. In such case, we approximate every physical quantity on an individual mesh which moreover is equipped with an independent hp-adaptivity mechanism. The idea of using different meshes is not new, but so far it has been used mainly in the context of various operator-splitting schemes. However, operator splitting techniques can destroy a lot of valuable information, and in particular information useful for error estimation and automatic adaptivity. Therefore we focus on monolithically coupled models. In this talk, we give a brief description of the multi-mesh hp-FEM, show applications to selected coupled problems, and discuss recent progress in open problems related to error estimation and automatic hp-adaptivity. We also present a new approach to automatic hp-adaptivity for time-dependent problems on dynamical hp-meshes. Numerous examples and applications are shown.

Some references: [1] P. Solin, L. Dubcova, I. Dolezel: On Adaptive hp-FEM With Arbitrary-Level Hanging Nodes for Approximations in H(curl), submitted to J. Comput. Appl. Math., May 2008. [2] P. Solin, J. Cerveny, I. Dolezel: Arbitrary-Level Hanging Nodes and Automatic Adaptivity in the hp-FEM, Math. Comput. Simul 77 (2008), 117 - 132. [3] P. Solin, J. Cerveny, L. Dubcova: Adaptive Multi-Mesh $hp$-FEM for Linear Thermoelasticity, submitted to CMAME, November 2007. [4] P. Solin, J. Cerveny, L. Dubcova, I. Dolezel: Multi-Mesh hp-FEM for Thermally Conductive Incompressible Flow. In: Proceedings of ECCOMAS Conference COUPLED PROBLEMS 2007 (M. Papadrakakis, E. Onate, B. Schrefler Eds.), CIMNE, Barcelona.