@article{55459, author = {{Bullerjahn, Nils and Kovács, Balázs}}, journal = {{IMA Journal of Numerical Analysis}}, title = {{{Error estimates for full discretization of Cahn--Hilliard equation with dynamic boundary conditions}}}, doi = {{10.1093/imanum/draf009}}, year = {{2025}}, } @article{53141, author = {{Edelmann, Dominik and Kovács, Balázs and Lubich, Christian}}, journal = {{IMA Journal of Numerical Analysis}}, number = {{5}}, pages = {{2581----2627}}, title = {{{Numerical analysis of an evolving bulk--surface model of tumour growth}}}, doi = {{10.1093/imanum/drae077}}, volume = {{45}}, year = {{2025}}, } @article{55781, abstract = {{In this paper, we prove that spatially semi-discrete evolving finite element method for parabolic equations on a given evolving hypersurface of arbitrary dimensions preserves the maximal $L^p$-regularity at the discrete level. We first establish the results on a stationary surface and then extend them, via a perturbation argument, to the case where the underlying surface is evolving under a prescribed velocity field. The proof combines techniques in evolving finite element method, properties of Green's functions on (discretised) closed surfaces, and local energy estimates for finite element methods}}, author = {{Bai, Genming and Kovács, Balázs and Li, Buyang}}, journal = {{IMA Journal of Numerical Analysis}}, title = {{{Maximal regularity of evolving FEMs for parabolic equations on an evolving surface}}}, doi = {{10.1093/imanum/draf082.}}, year = {{2025}}, } @article{45972, author = {{Kovács, Balázs}}, journal = {{SIAM Journal on Scientific Computing}}, number = {{2}}, pages = {{A645----A669}}, title = {{{Numerical surgery for mean curvature flow of surfaces}}}, doi = {{10.1137/22M1531919}}, volume = {{46}}, year = {{2024}}, } @unpublished{55078, abstract = {{This paper develops and discusses a residual-based a posteriori error estimate and a space--time adaptive algorithm for solving parabolic surface partial differential equations on closed stationary surfaces. The full discretization uses the surface finite element method in space and the backward Euler method in time. The proposed error indicator bounds the error quantities globally in space from above and below, and globally in time from above and locally from below. A space--time adaptive algorithm is proposed using the derived error indicator. Numerical experiments illustrate and complement the theory.}}, author = {{Kovács, Balázs and Lantelme, Michael Frederik Raúl}}, booktitle = {{arXiv:2407.02101}}, title = {{{A posteriori error estimates for parabolic partial differential equations on stationary surfaces}}}, year = {{2024}}, } @article{45971, abstract = {{Abstract An error estimate for a canonical discretization of the harmonic map heat flow into spheres is derived. The numerical scheme uses standard finite elements with a nodal treatment of linearized unit-length constraints. The analysis is based on elementary approximation results and only uses the discrete weak formulation.}}, author = {{Bartels, Sören and Kovács, Balázs and Wang, Zhangxian}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints}}}, doi = {{10.1093/imanum/drad037}}, year = {{2023}}, } @article{53140, abstract = {{We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fernández [\textit{SIAM J. Numer. Anal.}, 47 (2009), pp. 409-439] and provide a unified theoretical analysis of backward difference formulae (BDF methods) of order 1 to 6. The main novelty of our approach lies in the use of Dahlquist's G-stability concept together with multiplier techniques introduced by Nevannlina-Odeh and recently by Akrivis et al. [\textit{SIAM J. Numer. Anal.}, 59 (2021), pp. 2449-2472] to derive optimal stability and error estimates for both the velocity and the pressure. When combined with a method dependent Ritz projection for the initial data, unconditional stability can be shown while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations.}}, author = {{Contri, Alessandro and Kovács, Balázs and Massing, André}}, journal = {{arXiv}}, title = {{{Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates}}}, doi = {{10.48550/ARXIV.2312.05511}}, year = {{2023}}, } @article{45970, abstract = {{ We introduce a new phase field model for tumor growth where viscoelastic effects are taken into account. The model is derived from basic thermodynamical principles and consists of a convected Cahn–Hilliard equation with source terms for the tumor cells and a convected reaction–diffusion equation with boundary supply for the nutrient. Chemotactic terms, which are essential for the invasive behavior of tumors, are taken into account. The model is completed by a viscoelastic system consisting of the Navier–Stokes equation for the hydrodynamic quantities, and a general constitutive equation with stress relaxation for the left Cauchy–Green tensor associated with the elastic part of the total mechanical response of the viscoelastic material. For a specific choice of the elastic energy density and with an additional dissipative term accounting for stress diffusion, we prove existence of global-in-time weak solutions of the viscoelastic model for tumor growth in two space dimensions [Formula: see text] by the passage to the limit in a fully-discrete finite element scheme where a CFL condition, i.e. [Formula: see text], is required. Moreover, in arbitrary dimensions [Formula: see text], we show stability and existence of solutions for the fully-discrete finite element scheme, where positive definiteness of the discrete Cauchy–Green tensor is proved with a regularization technique that was first introduced by Barrett and Boyaval [Existence and approximation of a (regularized) Oldroyd-B model, Math. Models Methods Appl. Sci. 21 (2011) 1783–1837]. After that, we improve the regularity results in arbitrary dimensions [Formula: see text] and in two dimensions [Formula: see text], where a CFL condition is required. Then, in two dimensions [Formula: see text], we pass to the limit in the discretization parameters and show that subsequences of discrete solutions converge to a global-in-time weak solution. Finally, we present numerical results in two dimensions [Formula: see text]. }}, author = {{Garcke, Harald and Kovács, Balázs and Trautwein, Dennis}}, issn = {{0218-2025}}, journal = {{Mathematical Models and Methods in Applied Sciences}}, keywords = {{Applied Mathematics, Modeling and Simulation}}, number = {{13}}, pages = {{2673--2758}}, publisher = {{World Scientific Pub Co Pte Ltd}}, title = {{{Viscoelastic Cahn–Hilliard models for tumor growth}}}, doi = {{10.1142/s0218202522500634}}, volume = {{32}}, year = {{2022}}, } @article{45969, abstract = {{AbstractAn evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction–diffusion process on the surface, inspired by a gradient flow of a coupled energy. Two algorithms are proposed, both based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. One of the numerical methods is proved to be convergent in the$$H^1$$H1norm with optimal-order for finite elements of degree at least two. We present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.}}, author = {{Elliott, Charles M. and Garcke, Harald and Kovács, Balázs}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{4}}, pages = {{873--925}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Numerical analysis for the interaction of mean curvature flow and diffusion on closed surfaces}}}, doi = {{10.1007/s00211-022-01301-3}}, volume = {{151}}, year = {{2022}}, } @article{45963, abstract = {{AbstractThe scattering of electromagnetic waves from obstacles with wave-material interaction in thin layers on the surface is described by generalized impedance boundary conditions, which provide effective approximate models. In particular, this includes a thin coating around a perfect conductor and the skin effect of a highly conducting material. The approach taken in this work is to derive, analyse and discretize a system of time-dependent boundary integral equations that determines the tangential traces of the scattered electric and magnetic fields. In a familiar second step, the fields are evaluated in the exterior domain by a representation formula, which uses the time-dependent potential operators of Maxwell’s equations. The time-dependent boundary integral equation is discretized with Runge–Kutta based convolution quadrature in time and Raviart–Thomas boundary elements in space. Using the frequency-explicit bounds from the well-posedness analysis given here together with known approximation properties of the numerical methods, the full discretization is proved to be stable and convergent, with explicitly given rates in the case of sufficient regularity. Taking the same Runge–Kutta based convolution quadrature for discretizing the time-dependent representation formulas, the optimal order of convergence is obtained away from the scattering boundary, whereas an order reduction occurs close to the boundary. The theoretical results are illustrated by numerical experiments.}}, author = {{Nick, Jörg and Kovács, Balázs and Lubich, Christian}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{4}}, pages = {{1123--1164}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Time-dependent electromagnetic scattering from thin layers}}}, doi = {{10.1007/s00211-022-01277-0}}, volume = {{150}}, year = {{2022}}, } @article{45964, abstract = {{Abstract Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is shown that backward difference time discretizations of linear parabolic equations on an evolving surface along characteristic trajectories can preserve maximal $L^p$-regularity in the discrete setting. The result is applied to prove the stability and convergence of time discretizations of nonlinear parabolic equations on an evolving surface, with linearly implicit backward differentiation formulae characteristic trajectories of the surface, for general locally Lipschitz nonlinearities. The discrete maximal $L^p$-regularity is used to prove the boundedness and stability of numerical solutions in the $L^\infty (0,T;W^{1,\infty })$ norm, which is used to bound the nonlinear terms in the stability analysis. Optimal-order error estimates of time discretizations in the $L^\infty (0,T;W^{1,\infty })$ norm is obtained by combining the stability analysis with the consistency estimates.}}, author = {{Kovács, Balázs and Li, Buyang}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems}}}, doi = {{10.1093/imanum/drac033}}, year = {{2022}}, } @article{45966, abstract = {{Abstract This paper studies bulk–surface splitting methods of first order for (semilinear) parabolic partial differential equations with dynamic boundary conditions. The proposed Lie splitting scheme is based on a reformulation of the problem as a coupled partial differential–algebraic equation system, i.e., the boundary conditions are considered as a second dynamic equation that is coupled to the bulk problem. The splitting approach is combined with bulk–surface finite elements and an implicit Euler discretization of the two subsystems. We prove first-order convergence of the resulting fully discrete scheme in the presence of a weak CFL condition of the form $\tau \leqslant c h$ for some constant $c>0$. The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type.}}, author = {{Altmann, Robert and Kovács, Balázs and Zimmer, Christoph}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, number = {{2}}, pages = {{950--975}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Bulk–surface Lie splitting for parabolic problems with dynamic boundary conditions}}}, doi = {{10.1093/imanum/drac002}}, volume = {{43}}, year = {{2022}}, } @article{45968, abstract = {{Abstract We derive a numerical method, based on operator splitting, to abstract parabolic semilinear boundary coupled systems. The method decouples the linear components that describe the coupling and the dynamics in the abstract bulk- and surface-spaces, and treats the nonlinear terms similarly to an exponential integrator. The convergence proof is based on estimates for a recursive formulation of the error, using the parabolic smoothing property of analytic semigroups, and a careful comparison of the exact and approximate flows. This analysis also requires a deep understanding of the effects of the Dirichlet operator (the abstract version of the harmonic extension operator), which is essential for the stable coupling in our method. Numerical experiments, including problems with dynamic boundary conditions, reporting on convergence rates are presented.}}, author = {{Csomós, Petra and Farkas, Bálint and Kovács, Balázs}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Error estimates for a splitting integrator for abstract semilinear boundary coupled systems}}}, doi = {{10.1093/imanum/drac079}}, year = {{2022}}, } @article{45958, abstract = {{AbstractIn this paper, we consider a non-linear fourth-order evolution equation of Cahn–Hilliard-type on evolving surfaces with prescribed velocity, where the non-linear terms are only assumed to have locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix–vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. The paper is concluded by a variety of numerical experiments.}}, author = {{Beschle, Cedric Aaron and Kovács, Balázs}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{1}}, pages = {{1--48}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Stability and error estimates for non-linear Cahn–Hilliard-type equations on evolving surfaces}}}, doi = {{10.1007/s00211-022-01280-5}}, volume = {{151}}, year = {{2022}}, } @article{45956, abstract = {{Abstract The full Maxwell equations in the unbounded three-dimensional space coupled to the Landau–Lifshitz–Gilbert equation serve as a well-tested model for ferromagnetic materials. We propose a weak formulation of the coupled system based on the boundary integral formulation of the exterior Maxwell equations. We show existence and partial uniqueness of a weak solution and propose a new numerical algorithm based on finite elements and boundary elements as spatial discretization with backward Euler and convolution quadrature for the time domain. This is the first numerical algorithm which is able to deal with the coupled system of Landau–Lifshitz–Gilbert equation and full Maxwell’s equations without any simplifications like quasi-static approximations (e.g. eddy current model) and without restrictions on the shape of the domain (e.g. convexity). We show well-posedness and convergence of the numerical algorithm under minimal assumptions on the regularity of the solution. This is particularly important as there are few regularity results available and one generally expects the solution to be non-smooth. Numerical experiments illustrate and expand on the theoretical results.}}, author = {{Bohn, Jan and Feischl, Michael and Kovács, Balázs}}, issn = {{1609-4840}}, journal = {{Computational Methods in Applied Mathematics}}, keywords = {{Applied Mathematics, Computational Mathematics, Numerical Analysis}}, number = {{1}}, pages = {{19--48}}, publisher = {{Walter de Gruyter GmbH}}, title = {{{FEM-BEM Coupling for the Maxwell–Landau–Lifshitz–Gilbert Equations via Convolution Quadrature: Weak Form and Numerical Approximation}}}, doi = {{10.1515/cmam-2022-0145}}, volume = {{23}}, year = {{2022}}, } @article{45967, author = {{Binz, Tim and Kovács, Balázs}}, journal = {{arXiv}}, title = {{{A convergent finite element algorithm for mean curvature flow in higher codimension}}}, year = {{2021}}, } @article{45962, abstract = {{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.}}, author = {{Binz, Tim and Kovács, Balázs}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, number = {{3}}, pages = {{2545--2588}}, publisher = {{Oxford University Press (OUP)}}, title = {{{A convergent finite element algorithm for generalized mean curvature flows of closed surfaces}}}, doi = {{10.1093/imanum/drab043}}, volume = {{42}}, year = {{2021}}, } @article{45957, abstract = {{AbstractA 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.}}, author = {{Harder, Paula and Kovács, Balázs}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, number = {{3}}, pages = {{2589--2620}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Error estimates for the Cahn–Hilliard equation with dynamic boundary conditions}}}, doi = {{10.1093/imanum/drab045}}, volume = {{42}}, year = {{2021}}, } @article{45961, author = {{Nick, Jörg and Kovács, Balázs and Lubich, Christian}}, issn = {{0029-599X}}, journal = {{Numerische Mathematik}}, keywords = {{Applied Mathematics, Computational Mathematics}}, number = {{4}}, pages = {{997--1000}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Correction to: Stable and convergent fully discrete interior–exterior coupling of Maxwell’s equations}}}, doi = {{10.1007/s00211-021-01196-6}}, volume = {{147}}, year = {{2021}}, } @article{45953, abstract = {{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.}}, author = {{Hipp, David and Kovács, Balázs}}, issn = {{0272-4979}}, journal = {{IMA Journal of Numerical Analysis}}, keywords = {{Applied Mathematics, Computational Mathematics, General Mathematics}}, number = {{1}}, pages = {{638--728}}, publisher = {{Oxford University Press (OUP)}}, title = {{{Finite element error analysis of wave equations with dynamic boundary conditions: L2 estimates}}}, doi = {{10.1093/imanum/drz073}}, volume = {{41}}, year = {{2020}}, }