A stabilized trace finite element method for partial differential equations on evolving surfaces
Lehrenfeld C., Ol`shanskij M. A., Xu X.
SIAM Journal on Numerical Analysis
Vol.56, Issue3, P. 1643-1672
Опубликовано: 2018
Тип ресурса: Статья
Аннотация:
In this paper, we study a new numerical method for the solution of partial differential equations on evolving surfaces. The numerical method is built on the stabilized trace finite element method (TraceFEM) for the spatial discretization and finite differences for the time discretization. The TraceFEM uses a stationary background mesh, which can be chosen independent of time and the position of the surface. The stabilization ensures well-conditioning of the algebraic systems and defines a regular extension of the solution from the surface to its volumetric neighborhood. Having such an extension is essential for the numerical method to be well defined. The paper proves numerical stability and optimal order error estimates for the case of simplicial background meshes and finite element spaces of order m \geq 1. For the algebraic condition numbers of the resulting systems we prove estimates, which are independent of the position of the interface. The method allows that the surface and its
Ключевые слова:
Evolving surfaces; Level set method; Surface PDEs; TraceFEM
Algebra; Finite element method; Number theory; Partial differential equations; Trace elements; Algebraic conditions; Finite element space; Level Set method; Numerical experiments; Optimal order error estimates; Spatial discretizations; Time discretization; TraceFEM; Numerical methods
Язык текста: Английский
ISSN: 1095-7170
Lehrenfeld C.
Ol`shanskij M. A. Maksim Aleksandrovich 1971-
Xu X.
Лехренфелд C.
Ольшанский М. А. Максим Александрович 1971-
Ху Х.
A stabilized trace finite element method for partial differential equations on evolving surfaces
Текст визуальный непосредственный
SIAM Journal on Numerical Analysis
Society for Industrial and Applied Mathematics
Vol.56, Issue3 P. 1643-1672
2018
Статья
Evolving surfaces Level set method Surface PDEs TraceFEM
Algebra Finite element method Number theory Partial differential equations Trace elements Algebraic conditions Finite element space Level Set method Numerical experiments Optimal order error estimates Spatial discretizations Time discretization TraceFEM Numerical methods
In this paper, we study a new numerical method for the solution of partial differential equations on evolving surfaces. The numerical method is built on the stabilized trace finite element method (TraceFEM) for the spatial discretization and finite differences for the time discretization. The TraceFEM uses a stationary background mesh, which can be chosen independent of time and the position of the surface. The stabilization ensures well-conditioning of the algebraic systems and defines a regular extension of the solution from the surface to its volumetric neighborhood. Having such an extension is essential for the numerical method to be well defined. The paper proves numerical stability and optimal order error estimates for the case of simplicial background meshes and finite element spaces of order m \geq 1. For the algebraic condition numbers of the resulting systems we prove estimates, which are independent of the position of the interface. The method allows that the surface and its