# Higher-order time discretizations with ALE finite elements for parabolic problems on evolving surfaces

@article{Kovacs2014HigherorderTD, title={Higher-order time discretizations with ALE finite elements for parabolic problems on evolving surfaces}, author={Bal'azs Kov'acs and C. Guerra}, journal={arXiv: Numerical Analysis}, year={2014} }

A linear evolving surface partial differential equation is first discretized in space by an arbitrary Lagrangian Eulerian (ALE) evolving surface finite element method, and then in time either by a Runge-Kutta method, or by a backward difference formula. The ALE technique allows to maintain the mesh regularity during the time integration, which is not possible in the original evolving surface finite element method. Unconditional stability and optimal order convergence of the full discretizations… Expand

#### 9 Citations

High-order evolving surface finite element method for parabolic problems on evolving surfaces

- Mathematics
- 2016

High-order spatial discretisations and full discretisations of parabolic partial differential equations on evolving surfaces are studied. We prove convergence of the high-order evolving surface… Expand

Convergence of finite elements on an evolving surface driven by diffusion on the surface

- Mathematics, Computer Science
- Numerische Mathematik
- 2017

A novel stability and convergence analysis for evolving surface finite elements for the coupled problem of surface diffusion and surface evolution is developed and works with the matrix–vector formulation of the method and does not use geometric arguments. Expand

Computing arbitrary Lagrangian Eulerian maps for evolving surfaces

- Mathematics
- 2016

The good mesh quality of an evolving discretized surface or domain is often compromised during time evolution. In recent years this phenomena have been overcome in a couple of ways, one of them uses… Expand

A unified theory for continuous-in-time evolving finite element space approximations to partial differential equations in evolving domains

- Mathematics
- 2017

We develop a unified theory for continuous in time finite element discretisations of partial differential equations posed in evolving domains including the consideration of equations posed on… Expand

Numerical Analysis of the evolving surface finite element method for some parabolic problems

- Physics
- 2017

The present work investigates the evolving surface finite element method (ESFEM). One of its many applications is to approximate the solution of a parabolic partial differential equation on an… Expand

A convergent algorithm for mean curvature flow driven by diffusion on the surface

- Mathematics, Physics
- ArXiv
- 2019

Numerical examples are provided to support and complement the theoretical convergence results (demonstrating the convergence properties of the method without error estimate), and demonstrate the effectiveness of the methods in simulating a three-dimensional tumour growth model. Expand

Maximum norm stability and error estimates for the evolving surface finite element method

- Mathematics
- 2015

We show convergence in the natural L ∞ and W 1 , ∞ norm for a semidiscretization with linear finite elements of a linear parabolic partial differential equations on evolving surfaces. To prove this,… Expand

Error estimates for general non-linear Cahn-Hilliard equations on evolving surfaces

- Mathematics, Computer Science
- ArXiv
- 2020

The anti-symmetric structure of the weak equation system is preserved by the matrix-vector formulation and it is utilised to prove optimal-order and uniform-in-time error estimates. Expand

Error estimates for the Cahn-Hilliard equation with dynamic boundary conditions

- Computer Science, Mathematics
- ArXiv
- 2020

A proof of convergence is given for bulk--surface finite element semi-discretisation of the Cahn--Hilliard equation with Cahn--Hilliard-type dynamic boundary conditions in a smooth domain. The… Expand

#### References

SHOWING 1-10 OF 31 REFERENCES

Runge–Kutta time discretization of parabolic differential equations on evolving surfaces

- Mathematics
- 2012

A linear parabolic differential equation on a moving surface is first discretized in space by evolving surface finite elements and then in time by an implicit Runge–Kutta method. For algebraically… Expand

Gauss–Runge–Kutta time discretization of wave equations on evolving surfaces

- Mathematics, Computer Science
- Numerische Mathematik
- 2015

Under sufficient regularity conditions, optimal-order error estimates for this class of fully discrete methods are shown and some of the theoretical results are presented, aiming for higher-order accuracy in time and unconditional stability of the fully discrete scheme. Expand

Backward difference time discretization of parabolic differential equations on evolving surfaces

- Mathematics
- 2013

A linear parabolic differential equation on a moving surface is discretized in space by evolving surface finite elements and in time by backward difference formulas (BDF). Using results from… Expand

A Fully Discrete Evolving Surface Finite Element Method

- Mathematics, Computer Science
- SIAM J. Numer. Anal.
- 2012

This work proves optimal order error bounds for a backward Euler time discretization for linear finite elements. Expand

VARIATIONAL DISCRETIZATION OF LINEAR WAVE EQUATIONS ON EVOLVING SURFACES

- 2012

A linear wave equation on a moving surface is derived from Hamilton’s principle of stationary action. The variational principle is discretized with functions that are piecewise linear in space and… Expand

Time-discrete higher order ALE formulations: a priori error analysis

- Mathematics, Computer Science
- Numerische Mathematik
- 2013

We derive optimal a priori error estimates for discontinuous Galerkin (dG) time discrete schemes of any order applied to an advection–diffusion model defined on moving domains and written in the… Expand

An ALE ESFEM for Solving PDEs on Evolving Surfaces

- Mathematics
- 2012

Numerical methods for approximating the solution of partial differential equations on evolving hypersurfaces using surface finite elements on evolving triangulated surfaces are presented. In the ALE… Expand

Variational discretization of wave equations on evolving surfaces

- Computer Science, Mathematics
- Math. Comput.
- 2015

This work studies stability and convergence of the full discretization in the natural time-dependent norms under the same CFL condition that is required for a fixed surface and proves optimal-order error bounds. Expand

Numerical Analysis of Partial Differential Equations on Evolving Surfaces

- Mathematics
- 2013

This dissertation addresses the numerical study of full discretization methods for linear parabolic equations as well as wave equations on evolving surfaces. It is the first work able to give… Expand

Time-Discrete Higher-Order ALE Formulations: Stability

- Mathematics, Computer Science
- SIAM J. Numer. Anal.
- 2013

This work proposes time-discrete discontinuous Galerkin (dG) numerical schemes of any order for a time-dependent advection-diffusion-model problem in moving domains, and proves that conservative and nonconservative dG schemes are equivalent and unconditionally stable. Expand