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The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion.The physical domain is embedded in a triangulated computational domain and can overlap the time-independent background mesh in an arbitrary way. The numerical method is based on finite difference discretizations of time derivatives and a standard geometrically unfitted finite element method with an additional stabilization term in the spatial domain.The performance and analysis of the method rely on the fundamental extension result in Sobolev spaces for functions defined on bounded domains. This paper includes a complete stability and error analysis, which accounts for discretization errors resulting from finite difference and finite element approximations as well as for geometric errors coming from a possible approximate recovery of the physical domain. Several numerical examples illustrate the theory and demonstrate the practical efficiency of the method.
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This content will become publicly available on June 25, 2026
A Conservative Eulerian Finite Element Method for Transport and Diffusion in Moving Domains
Abstract The paper introduces a finite element method for an Eulerian formulation of partial differential equations governing the transport and diffusion of a scalar quantity in a time-dependent domain. The method follows the idea from[C. Lehrenfeld and M. Olshanskii,An Eulerian finite element method for PDEs in time-dependent domains,ESAIM Math. Model. Numer. Anal. 53 2019, 2, 585–614]of a solution extension to realise the Eulerian time-stepping scheme. However, a reformulation of the partial differential equation is suggested to derive a scheme which conserves the quantity under consideration exactly on the discrete level. For the spatial discretisation, the paper considers an unfitted finite element method. Ghost-penalty stabilisation is used to realise the discrete solution extension and gives a scheme robust against arbitrary intersections between the mesh and geometry interface. The stability is analysed for both first- and second-order backward differentiation formula versions of the scheme. Several numerical examples in two and three spatial dimensions are included to illustrate the potential of this method.
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- Award ID(s):
- 1929284
- PAR ID:
- 10617028
- Publisher / Repository:
- De Gruyter Brill
- Date Published:
- Journal Name:
- Computational Methods in Applied Mathematics
- ISSN:
- 1609-4840
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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