RUI: Efficient Numerical Methods for Axisymmetric Problems

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An axisymmetric problem is a problem defined on a three-dimensional (3D) domain that is symmetric with respect to an axis. These problems arise in various applications in the field of biomedical engineering, electromagnetism, and optics. An axisymmetric problem can be reduced to a sequence of two-dimensional (2D) problems by using cylindrical coordinates and a Fourier series decomposition. A discrete problem corresponding to a 2D problem is significantly smaller than a discrete problem corresponding to a 3D one, so such dimension reduction is an attractive feature considering computation time. The resulting 2D problem, however, is posed in weighted function spaces and is mathematically quite different from the analogous "standard" 2D problems, so special care is required when developing numerical methods that are well-fit for these weighted 2D problems. In this project, we will study efficient numerical techniques with solid mathematical support that can be applied to axisymmetric problems including those that arise in the treatment of various cancer treatments. The first goal of this project is to perform multigrid analysis for axisymmetric H(curl) and H(div) problems with general data including the axisymmetric time harmonic Maxwell equations. Multigrid for axisymmetric H(curl) and H(div) problems have been studied previously under the assumption that the data is independent of the rotational variable, which is not the case for most applications. Therefore, this project will bring new results for general axisymmetric problems with meaningful applications in Hepatic Microwave Ablation, an alternate treatment to liver, breast, bone, and lung cancer. Undergraduate students will be a part of evaluating the performance of multigrid in designing efficient antennas that can be used for these cancer treatments. Furthermore, this project will provide new mathematical tools to study axisymmetric problems with general data as well. The second goal of this project is to study axisymmetric state-constrained elliptic optimal control problems with axisymmetric data by using P1 finite element methods. There are very few studies done on axisymmetric optimal control problems, so this will be new and significant. The PI will also run a "Numerical Analysis Day" for local high school students with her undergraduate students from the James Madison University Association for Women in Mathematics Student Chapter.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.