Ganghui Zhang: Structure-preserving parametric finite element method for surface diffusion based on Lagrange multiplier approaches

Vortragender/Speaker: Ganghui Zhang (Yau Mathematical Sciences Center, Tsinghua University)
Ansprechpartner/Contact: Dr. Simon Praetorius

Titel/Title:
Structure-preserving parametric finite element method for surface diffusion based on Lagrange multiplier approaches

Zusammenfassung/Abstract:
In this talk, we discuss a novel formulation for parametric finite element methods
to simulate surface diffusion for both closed curves and surfaces. Several high-order
temporal discretizations are proposed based on this new formulation. To ensure that
the numerical methods preserve geometric structures of surface diffusion, our formula-
tion incorporates two scalar Lagrange multipliers and two evolution equations involving
the perimeter/surface-area and area/volume, respectively. By discretizing the spatial
variable using piecewise linear finite elements and the temporal variable using the back-
ward differentiation formulae method, we develop first-order and second-order tempo-
ral schemes that effectively preserve the structure at a fully discrete level. These new
schemes are implicit and can be efficiently solved using Newton’s method. Extensive
numerical experiments demonstrate that our methods achieve the desired temporal ac-
curacy, while simultaneously preserving the geometric structure of the surface diffusion.