This proposal considers innovative numerical methods using hybridization and multigrid techniques. While hybridization was thought of as an implementation technique in its inception, the investigator argues that it is much more. It is shown to be a computational technique that has the potential of achieving goals difficult to achieve by conventional techniques in applications ranging from computational fluid mechanics to electromagnetics. E.g., a new technique using hybridization to solve a long standing research problem, namely computing exactly incompressible approximations to fluid flow, is proposed. Other proposed uses of hybridization include construction and analysis of variable degree mixed methods and design of local post-processing techniques to enhance accuracy. A different category of proposed activities involve accelerated solution techniques by multigrid methods. Powerful multigrid techniques are proposed for various systems arising from computational electromagnetics and hybridization. The aim is to make solution techniques efficient. This goal, while often achieved by multigrid methods, can also be achieved by utilizing symmetries such as axisymmetry occurring in practical problems. E.g., investigation of methods for the axisymmetric Maxwell equations form a part of the proposed activities.
While the proposed research has significant theoretical impact, equally important is the fact that parts of the research are motivated by the needs of industrial applications such as clinical ablation. Use of the proposed new methods in these applications is an integral part of the proposal. Broader impacts of the proposed activities result not only from the applications to the industrial problems, but also from educational activities and broad dissemination of results.
