A convergent finite element algorithm for generalized mean curvature flows of closed surfaces - Binz, Tim; Kovács, Balázs (IMA Journal of Numerical Analysis)
Abstract An algorithm is proposed for generalized mean curvature flow of closed two-dimensional surfaces, which include inverse mean curvature flow and powers of mean and inverse mean curvature flow. Error estimates are proved for semidiscretizations and full discretizations for the generalized flow. The algorithm proposed and studied here combines evolving surface finite elements, whose nodes determine the discrete surface, and linearly implicit backward difference formulae for time integration. The numerical method is based on a system coupling the surface evolution to nonlinear second-order parabolic evolution equations for the normal velocity and normal vector. A convergence proof is presented in the case of finite elements of polynomial degree at least 2 and backward difference formulae of orders 2 to 5. The error analysis combines stability estimates and consistency estimates to yield optimal-order $H^1$-norm error bounds for the computed surface position, velocity, normal vector, normal velocity and therefore for the mean curvature. The stability analysis is performed in the matrix–vector formulation and is independent of geometric arguments, which only enter the consistency analysis. Numerical experiments are presented to illustrate the convergence results and also to report on monotone quantities, e.g. Hawking mass for inverse mean curvature flow, and complemented by experiments for nonconvex surfaces.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Error estimates for the Cahn–Hilliard equation with dynamic boundary conditions - Harder, Paula; Kovács, Balázs (IMA Journal of Numerical Analysis)
Abstract A proof of convergence is given for a bulk–surface finite element semidiscretisation of the Cahn–Hilliard equation with Cahn–Hilliard-type dynamic boundary conditions in a smooth domain. The semidiscretisation is studied in an abstract weak formulation as a second-order system. Optimal-order uniform-in-time error estimates are shown in the $L^2$- and $H^1$-norms. The error estimates are based on a consistency and stability analysis. The proof of stability is performed in an abstract framework, based on energy estimates exploiting the anti-symmetric structure of the second-order system. Numerical experiments illustrate the theoretical results.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Correction to: Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations - Nick, Jörg; Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

A convergent evolving finite element algorithm for Willmore flow of closed surfaces - Kovács, Balázs; Li, Buyang; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Finite element error analysis of wave equations with dynamic boundary conditions: L2 estimates - Hipp, David; Kovács, Balázs (IMA Journal of Numerical Analysis)
Abstract $L^2$ norm error estimates of semi- and full discretizations of wave equations with dynamic boundary conditions, using bulk–surface finite elements and Runge–Kutta methods, are studied. The analysis rests on an abstract formulation and error estimates, via energy techniques, within this abstract setting. Four prototypical linear wave equations with dynamic boundary conditions are analysed, which fit into the abstract framework. For problems with velocity terms or with acoustic boundary conditions we prove surprising results: for such problems the spatial convergence order is shown to be less than 2. These can also be observed in the presented numerical experiments.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Finite element error analysis of wave equations with dynamic boundary conditions: L2 estimates - Hipp, David; Kovács, Balázs (IMA Journal of Numerical Analysis)
Abstract $L^2$ norm error estimates of semi- and full discretizations of wave equations with dynamic boundary conditions, using bulk–surface finite elements and Runge–Kutta methods, are studied. The analysis rests on an abstract formulation and error estimates, via energy techniques, within this abstract setting. Four prototypical linear wave equations with dynamic boundary conditions are analysed, which fit into the abstract framework. For problems with velocity terms or with acoustic boundary conditions we prove surprising results: for such problems the spatial convergence order is shown to be less than 2. These can also be observed in the presented numerical experiments.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Higher-order linearly implicit full discretization of the Landau–Lifshitz–Gilbert equation - Akrivis, Georgios; Feischl, Michael; Kovács, Balázs; Lubich, Christian (Mathematics of Computation)
Keywords: Applied Mathematics; Computational Mathematics; Algebra and Number Theory

A convergent algorithm for forced mean curvature flow driven by diffusion on the surface - Kovács, Balázs; Li, Buyang; Lubich, Christian (Interfaces and Free Boundaries)
Keywords: Applied Mathematics

A convergent evolving finite element algorithm for mean curvature flow of closed surfaces - Kovács, Balázs; Li, Buyang; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Numerical analysis of partial differential equations on and of evolving surfaces - Kovács, Balázs

Discrete maximum principles for nonlinear elliptic finite element problems on surfaces with boundary - Karátson, János; Kovács, Balázs; Korotov, Sergey (IMA Journal of Numerical Analysis)
Abstract The maximum principle forms an important qualitative property of second-order elliptic equations; therefore, its discrete analogues, the so-called discrete maximum principles (DMPs), have drawn much attention owing to their role in reinforcing the qualitative reliability of the given numerical scheme. In this paper DMPs are established for nonlinear finite element problems on surfaces with boundary, corresponding to the classical pointwise maximum principles on Riemannian manifolds in the spirit of Pucci & Serrin (2007, The Maximum Principle. Springer). Various real-life examples illustrate the scope of the results.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Discrete maximum principles for nonlinear elliptic finite element problems on surfaces with boundary - Karátson, János; Kovács, Balázs; Korotov, Sergey (IMA Journal of Numerical Analysis)
Abstract The maximum principle forms an important qualitative property of second-order elliptic equations; therefore, its discrete analogues, the so-called discrete maximum principles (DMPs), have drawn much attention owing to their role in reinforcing the qualitative reliability of the given numerical scheme. In this paper DMPs are established for nonlinear finite element problems on surfaces with boundary, corresponding to the classical pointwise maximum principles on Riemannian manifolds in the spirit of Pucci & Serrin (2007, The Maximum Principle. Springer). Various real-life examples illustrate the scope of the results.
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Linearly implicit full discretization of surface evolution - Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Computing arbitrary Lagrangian Eulerian maps for evolving surfaces - Kovács, Balázs (Numerical Methods for Partial Differential Equations)
Keywords: Applied Mathematics; Computational Mathematics; Numerical Analysis; Analysis

Convergence of finite elements on an evolving surface driven by diffusion on the surface - Kovács, Balázs; Li, Buyang; Lubich, Christian; Power Guerra, Christian A. (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Stability and convergence of time discretizations of quasi-linear evolution equations of Kato type - Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations - Kovács, Balázs; Lubich, Christian (Numerische Mathematik)
Keywords: Applied Mathematics; Computational Mathematics

Maximum norm stability and error estimates for the evolving surface finite element method - Kovács, Balázs; Power Guerra, Christian Andreas (Numerical Methods for Partial Differential Equations)
Keywords: Applied Mathematics; Computational Mathematics; Numerical Analysis; Analysis

High-order evolving surface finite element method for parabolic problems on evolving surfaces - Kovács, Balázs (IMA Journal of Numerical Analysis)
Keywords: Applied Mathematics; Computational Mathematics; General Mathematics

Maximum norm stability and error estimates for the evolving surface finite element method - Kovács, Balázs; Power Guerra, Christian Andreas (Numerical Methods for Partial Differential Equations)
Keywords: Applied Mathematics; Computational Mathematics; Numerical Analysis; Analysis