2022 Fiscal Year Research-status Report
Surface PDE: a minimizing movement approach
| Project/Area Number |
22K03440
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| Research Institution | Meiji University |
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Principal Investigator |
Ginder Elliott 明治大学, 総合数理学部, 専任教授 (30648217)
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| Project Period (FY) |
2022-04-01 – 2025-03-31
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| Keywords | surface PDE / minimizing movements / interfacial dynamics / level set method / approximation methods |
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| Outline of Annual Research Achievements |
Our research focused on developing minimizing movments for surface-constrained partial differential equations. The corresponding approximation methods were successfully realized by incorporating the closest point method into a functional minimization scheme.
Using our surface-type minimizing movements, we were able to effectively simulate mean curvature flow and hyperbolic mean curvature flow of interfaces on surfaces. These methods represent generalizations of the MBO (Merriman, Bence, and Osher) and HMBO (Hyperbolic Merriman, Bence, and Osher) algorithms, specifically tailored for the surface-constrained setting.
In addition, we designed a surface-constrained signed distance vector field (SDVF) for describing phase geometries on surfaces in multiphase settings. We further implemented the numerical algorithms that enable the application of the SDVF to computational problems.
Regarding our approximation method that combines the closest point method and minimizing movements, numerical error analyses were conducted for the heat and wave equations on surfaces, under various conditions. Convergence of our surface-type minimizing movement, with respect to the spatial discretization, was also examined. Our the results revealed that the numerical solution converges to the exact solution.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Over the past year, our research on minimizing movements for surface partial differential equations has made significant progress, closely aligning with our original plan and encountering minimal obstacles. Our primary goal was to extend the applicability of the closest point method by incorporating it into the setting of minimizing movements.
By addressing this task early on, we effectively initiated our research and established momentum, allowing it to commence smoothly and efficiently. Notably, numerical tests and analyses played a vital role in confirming and bolstering our findings, providing robust evidence to support our research outcomes.
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| Strategy for Future Research Activity |
Looking ahead, the roadmap for our research included various tasks aimed at enhancing and refining our approximation methods. First of all, we endeavor to reduce the computational time of our methods--efficient, and accurate algorithms will enable new simulations and applications. Second, we will perform a numerical analysis of area preserving motions. This analysis will allow us to quantify and understand the numerical errors associated with our approximation methods, and to facilitate their refinement. Finally, it is necessary to evaluate the performance of our methods on complex geometries. These tests will access the robustness, stability, and accuracy of our methods on complex surface geometries. In turn, this will help to identify limitations and areas for improvement.
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| Causes of Carryover |
Funds allotted for travel were preserved due to partaking in online discussions. Transferred funds will be used to supplement travel plans.
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