Adaptive finite element methods for multi-scale/physics problems

Participants

• Axel Målqvist
• Donald Estep, Colorado State University
• Victor Ginting, University of Wyoming
• Michael Holst, University of California at San Diego
• Mats G. Larson, Umeå University
• Robert Söderlund, Umeå University
• Daniel Elfverson, Uppsala University

Research

Engineering applications that involves different kinds of physics (multi-physics) and different scales (multi-scale) are very computationally challanging. As computers get more and more powerful these problems can now be atttacked.

Multi-physics

Complicated coupled systems of non-linear partial differential equations can be solved on parallel machines. As a result of this development much more complex physical phenomena can be analyzed by computational means. Reliability and efficiency becomes crucial when solving these problems. Reliability comes down to a need for bounds of the error between the computed approximation and the exact solution, and efficiency is achieved by designing adaptive algorithms that distributes the computational effort where it is most needed.

Multi-Scale

Multiscale problems are some of the greatest challenges in computational mathematics today. In all branches of the engineering sciences we encounter problems with features on several different scales. A typical example is simulations in a heterogenous media where material data such as module of elasticity, conductivity or permeability, varies in space over several different scales. In order to solve these problems efficiently we propose an adaptive multiscale method where the critical parameters of the method are choosen automatically through an adaptive algorithm. We also study uncertainty in the multiscale data using non-parametric density estimation.

This project is supported by the Göran Gustafsson foundation and VR.

Publications

Selected publications related to the project.

• Convergence analysis of finite element approximations of the Joule heating problem in three spatial dimensions. Michael J. Holst, Mats G. Larson, Axel Målqvist, and Robert Söderlund. In BIT Numerical Mathematics, volume 50, pp 781-795, 2010. (DOI).
• Nonparametric density estimation for randomly perturbed elliptic problems II: Applications and adaptive modeling. Donald Estep, Axel Målqvist, and Simon Tavener. In International Journal for Numerical Methods in Engineering, volume 80, pp 846-867, 2009. (DOI).
• Nonparametric density estimation for randomly perturbed elliptic problems I: Computational methods, a posteriori analysis, and adaptive error control. Donald Estep, Axel Målqvist, and Simon Tavener. In SIAM Journal on Scientific Computing, volume 31, pp 2935-2959, 2009. (DOI).
• A mixed adaptive variational multiscale method with applications in oil reservoir simulation. Mats G. Larson and Axel Målqvist. In Mathematical Models and Methods in Applied Sciences, volume 19, pp 1017-1042, 2009. (DOI).
• An adaptive variational multiscale method for convection-diffusion problems. Mats G. Larson and Axel Målqvist. In Communications in Numerical Methods in Engineering, volume 25, pp 65-79, 2009. (DOI).

• Goal oriented adaptivity for coupled flow and transport problems with applications in oil reservoir simulations. Mats G. Larson and Axel Målqvist. In Computer Methods in Applied Mechanics and Engineering, volume 196, pp 3546-3561, 2007.

• Adaptive variational multiscale methods based on a posteriori error estimation: Energy norm estimates for elliptic problems. Mats G. Larson and Axel Målqvist. In Computer Methods in Applied Mechanics and Engineering, volume 196, pp 2313-2324, 2007.

Axel's webpage