Conveners
Numerical Methods for Geometric PDEs: MS-3-1
- Buyang Li (The Hong Kong Polytechnic University)
Numerical Methods for Geometric PDEs: MS-3-2
- Robert Nürnberg
Description
Organisers: Buyang Li, Robert Nürnberg
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Shawn Walker (Louisiana State University)08/09/2026, 16:30Numerical Methods for Geometric PDEs
We present a framework for computing shape derivatives of boundary functionals discretized with unfitted finite element methods. The main idea is to replace the boundary functional by a narrow-band volumetric regularization prior to discretization. This yields an exact Fr'echet derivative of the resulting discrete functional with respect to the discrete level set function, without any...
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Evan Gawlik (Santa Clara University)08/09/2026, 17:00Numerical Methods for Geometric PDEs
If a simplicial triangulation is equipped with a piecewise smooth Riemannian metric that has single-valued tangential-tangential components on element interfaces, then there are various notions of curvature that one can define, even though the classical formulas for curvature involving derivatives of the metric no longer make sense. In this talk, I will explain the origins of these...
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Yakov Berchenko-Kogan (Florida Institute of Technology)08/09/2026, 17:30Numerical Methods for Geometric PDEs
Blow-up finite elements, developed jointly with Evan Gawlik, were motivated by a vexing problem when discretizing tangent vector fields on surfaces: For a discretized surface, the angles at vertices generally no longer sum to 360 degrees. As a result, it is not possible to construct a vector field approximation that is continuous within each element, tangent to the surface, and continuous...
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RONG TANG (The Hong Kong Polytechnic University)08/09/2026, 18:00Numerical Methods for Geometric PDEs
Parametric finite element methods for curvature flows often add an artificial tangential motion to improve the quality of the evolving mesh. We first present a convergence proof for a parametric finite element method with the minimal deformation rate (MDR) tangential motion, established within a projected-distance framework without relying on auxiliary evolution equations for the mean...
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Paola Pozzi09/09/2026, 10:30Numerical Methods for Geometric PDEs
In this talk we discuss the question of finding a network configuration of minimal length connecting three given points in the Heisenberg group.
After formulating a suitable horizontal curve shortening flow, we present numerical experiments based on a stable fully discrete finite element scheme that provide useful insights into the rich landscape of this sub-Riemannian geometry.
This...
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Dr Quan Zhao (University of Science and Technology of China)09/09/2026, 11:00Numerical Methods for Geometric PDEs
We develop a constrained Onsager variational framework for parametric finite element approximations of Willmore and Helfrich flows. By incorporating the weak curvature relation as a PDE constraint, the bending energy variation is formulated in terms of the curvature vector, while a relaxed minimal-deformation-rate constraint determines the tangential velocity. The resulting mixed formulation...
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Carolin Mehlmann09/09/2026, 11:30Numerical Methods for Geometric PDEs
Nonconforming surface finite elements have recently been developed to discretize 3D vector-valued compressible flow problems arising in climate modeling. In this talk, we present an error analysis of this approach for a vector-valued Laplace problem, a key operator in fluid equations on surfaces. The problem is discretized via edge-integration on local flat triangles using the nonconforming...
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Dennis Trautwein (Universität Regensburg)09/09/2026, 12:00Numerical Methods for Geometric PDEs
The numerical simulation of viscoelastic two-phase flows involves complex free boundary dynamics and faces major challenges, such as the High Weissenberg Number Problem and the loss of positive definiteness of the conformation tensor.
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In this presentation, we introduce an energy-stable numerical framework designed to address these issues. First, we discuss energy-stable, positivity-preserving...