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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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Finite element approximation of the Isaacs equation
We propose and analyze a two-scale finite element method for the Isaacs equation. The fine scale is given by the mesh size h whereas the coarse scale ε is dictated by an integro-differential approximation of the partial differential equation. We show that the method satisfies the discrete maximum principle provided that the mesh is weakly acute. This, in conjunction with weak operator consistency of the finite element method, allows us to establish convergence of the numerical solution to the viscosity solution as ε , h → 0, and ε ≳ ( h |log h |) 1/2 . In addition, using a discrete Alexandrov Bakelman Pucci estimate we deduce rates of convergence, under suitable smoothness assumptions on the exact solution.
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- PAR ID:
- 10101483
- Date Published:
- Journal Name:
- ESAIM: Mathematical Modelling and Numerical Analysis
- Volume:
- 53
- Issue:
- 2
- ISSN:
- 0764-583X
- Page Range / eLocation ID:
- 351 to 374
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1051/m2an/2018067
