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Enriched Discontinuous Galerkin Methods for Highly Oscillatory Differential Equations

Schmitt, Kevin
http://rave.ohiolink.edu/etdc/view?acc_num=ucin1712915241789033

Abstract Details

2024, PhD, University of Cincinnati, Arts and Sciences: Mathematical Sciences.
This dissertation presents a comprehensive study of a novel enriched Discontinuous Galerkin (xDG) method, designed specifically for solving highly oscillatory differential equations, with a primary focus on its application to High-Intensity Focused Ultrasound (HIFU). HIFU is a medical procedure that employs ultrasound waves, ranging between 0.1 to 20 MHz, to target and ablate abnormal tissues within the body, serving as a motivating application for this research. Central to this study is a comparative analysis between the proposed enriched Symmetric Interior Penalty Galerkin (xSIPG) method and its predecessors, highlighting the advancements in solving crucial differential equations, notably the Helmholtz and Bioheat equations. These equations are pivotal for understanding the propagation of ultrasound waves and their interaction with human tissues. A significant achievement of this research is the optimization of penalty parameters within the xDG framework, which plays an essential role in the accuracy and computational efficiency of the methods. The results indicate xSIPG's significant improvement in modeling efficiency for HIFU simulations, potentially enhancing computational performance by up to three orders of magnitude compared to conventional FEM. Moreover, this study establishes the limitations of previous xDG methods in fully capturing the complexities of the HIFU model, a challenge addressed by the xSIPG approach. This advancement not only highlights the methodological leap facilitated by the xSIPG method but also reinforces the potential of applying xDG techniques to simulate HIFU.
Benjamin Vaughan, Ph.D. (Committee Chair)
Deniz Bilman, Ph.D. (Committee Member)
Sookkyung Lim, Ph.D. (Committee Member)
60 p.
Mathematics
discontinuous galerkin; oscillatory; numerical analysis; enriched method; computational; penalization

Recommended Citations

Citations

  • Schmitt, K. (2024). Enriched Discontinuous Galerkin Methods for Highly Oscillatory Differential Equations [Doctoral dissertation, University of Cincinnati]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1712915241789033

    APA Style (7th edition)

  • Schmitt, Kevin. Enriched Discontinuous Galerkin Methods for Highly Oscillatory Differential Equations. 2024. University of Cincinnati, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ucin1712915241789033.

    MLA Style (8th edition)

  • Schmitt, Kevin. "Enriched Discontinuous Galerkin Methods for Highly Oscillatory Differential Equations." Doctoral dissertation, University of Cincinnati, 2024. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1712915241789033

    Chicago Manual of Style (17th edition)

ucin1712915241789033
41
© 2024, some rights reserved.Creative Commons License Enriched Discontinuous Galerkin Methods for Highly Oscillatory Differential Equations by Kevin Schmitt is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by University of Cincinnati and OhioLINK.