A diffuse interface approach for vector-valued PDEs on surfaces

Axel Voigt (Institute of Scientific Computing, TU Dresden, Germany; and Center for Systems Biology, Dresden, Germanyand Cluster of Excellence Physics of Life (PoL), TU Dresden, Germany)

Abstract

Approximating PDEs on surfaces by the diffuse interface approach allows us to use standard numerical tools to solve these problems. This makes it an attractive numerical approach. We extend this approach to vector-valued surface PDEs and explore their convergence properties. In contrast to the well-studied case of scalar-valued surface PDEs, the optimal order of convergence can only be achieved if certain higher-order relations between mesh size and interface width are fulfilled. This difference results from the increased coupling between the surface geometry and the PDE for vector-valued quantities defined on it.

Keywords

surface PDEs, diffuse-interface approximation, finite-element approximation

2010 Mathematics Subject Classification

35K55, 35K65, 37E35

Full Text (PDF format)

Received 30 December 2023

Received revised 7 March 2024

Accepted 12 March 2024

Published 18 July 2024