Explanipedia

Analysis of a Schwarz-Fourier domain decomposition method Open

Arnold Reusken · 2025

The Schwarz domain decomposition method can be used for approximately solving a Laplace equation on a domain formed by the union of two overlapping discs. We consider an inexact variant of this method in which the subproblems on the discs …

A comparative study of finite element methods for a class of harmonic map heat flow problems Open

Nam H. Nguyen, Arnold Reusken · 2025

In this paper, we review and systematically compare three finite element discretization methods for a harmonic map heat flow problem from the unit disk in $\mathbb{R}^2$ to the unit sphere in $\mathbb{R}^3$ in an unified framework. Numeric…

Analysis of a finite element method for PDEs in evolving domains with topological changes Open

Maxim A. Olshanskii, Arnold Reusken · 2025

The paper presents the first rigorous error analysis of an unfitted finite element method for a linear parabolic problem posed on an evolving domain $Ω(t)$ that may undergo a topological change, such as, for example, a domain splitting. Th…

Discretization error analysis for a radially symmetric harmonic map heat flow problem Open

Nam H. Nguyen, Arnold Reusken · 2025

Supplementary material (source code) for the numerical experiments presented in "Discretization error analysis for a radially symmetric harmonic map heat flow problem" - Nguyen, Reusken (2025) "Error analysis for a finite element discretiz…

Numerical analysis of a constrained strain energy minimization problem Open

Tilman Aleman, Arnold Reusken · 2024

We consider a setting in which an evolving surface is implicitly characterized as the zero level of a level set function. Such an implicit surface does not encode any information about the path of a single point on the evolving surface. In…

A narrow band finite element method for the level set equation Open

Maxim A. Olshanskii, Arnold Reusken, Paul Schwering · 2024

A finite element method is introduced to track interface evolution governed by the level set equation. The method solves for the level set indicator function in a narrow band around the interface. An extension procedure, which is essential…

Analysis of the Taylor-Hood Surface Finite Element Method for the surface Stokes equation Open

Arnold Reusken · 2024

We consider the surface Stokes equation on a smooth closed hypersurface in three-dimensional space. For discretization of this problem a generalization of the surface finite element method (SFEM) of Dziuk-Elliott combined with a Hood-Taylo…

Analysis of a space-time unfitted finite element method for PDEs on evolving surfaces Open

Arnold Reusken, Hauke Sass · 2024

In this paper we analyze a space-time unfitted finite element method for the discretization of scalar surface partial differential equations on evolving surfaces. For higher order approximations of the evolving surface we use the technique…

Diffusion of tangential tensor fields: numerical issues and influence of geometric properties Open

Elena Bachini, PA Brandner, Thomas Jankuhn, Michael Nestler, Simon Praetorius , et al. · 2023

We study the diffusion of tangential tensor-valued data on curved surfaces. For this purpose, several finite-element-based numerical methods are collected and used to solve a tangential surface n -tensor heat flow problem. These methods di…

An Eulerian finite element method for tangential Navier-Stokes equations on evolving surfaces Open

Maxim A. Olshanskii, Arnold Reusken, Paul Schwering · 2023

The paper introduces a geometrically unfitted finite element method for the numerical solution of the tangential Navier--Stokes equations posed on a passively evolving smooth closed surface embedded in $\mathbb{R}^3$. The discrete formulat…

An Accurate and Robust Eulerian Finite Element Method for Partial Differential Equations on Evolving Surfaces Open

Hauke Sass, Arnold Reusken · 2022

In this paper we present a new Eulerian finite element method for the discretization of scalar partial differential equations on evolving surfaces. In this method we use the restriction of standard space-time finite element spaces on a fix…

Analysis of optimal preconditioners for CutFEM Open

Sven Groß, Arnold Reusken · 2022

In this article, we consider a class of unfitted finite element methods for scalar elliptic problems. These so‐called CutFEM methods use standard finite element spaces on a fixed unfitted triangulation combined with the Nitsche technique a…

Preconditioner experiments for 2D CutFEM Poisson Interface Problem Open

Sven Groß, Arnold Reusken · 2022

Python code used for numerical experiments in the article "Analysis of optimal preconditioners for CutFEM" by S. Gross and A. Reusken. Parameters can be set in the parameter section at the end of the file. The code is based on NGSolve toge…

On derivations of evolving surface Navier–Stokes equations Open

Philip Brandner, Arnold Reusken, Paul Schwering · 2022

In recent literature several derivations of incompressible Navier–Stokes-type equations that model the dynamics of an evolving fluidic surface have been presented. These derivations differ in the physical principles used in the modeling ap…

Finite Element Discretization Methods for Velocity-Pressure and Stream Function Formulations of Surface Stokes Equations Open

Philip Brandner, Thomas Jankuhn, Simon Praetorius, Arnold Reusken, Axel Voigt · 2022

In this paper we study parametric TraceFEM and parametric SurfaceFEM (SFEM) discretizations of a surface Stokes problem. These methods are applied both to the Stokes problem in velocity-pressure formulation and in stream function formulati…

Diffusion of tangential tensor fields: numerical issues and influence of geometric properties Open

Elena Bachini, Philip Brandner, Thomas Jankuhn, Michael Nestler, Simon Praetorius , et al. · 2022

We study the diffusion of tangential tensor-valued data on curved surfaces. For this purpose, several finite-element-based numerical methods are collected and used to solve a tangential surface n-tensor heat flow problem. These methods dif…

Tangential Navier-Stokes equations on evolving surfaces: Analysis and simulations Open

Maxim A. Olshanskii, Arnold Reusken, Alexander Zhiliakov · 2022

The paper considers a system of equations that models a lateral flow of a Boussinesq--Scriven fluid on a passively evolving surface embedded in $\mathbb{R}^3$. For the resulting Navier-Stokes type system, posed on a smooth closed time-depe…

On derivations of evolving surface Navier-Stokes equations Open

Philip Brandner, Arnold Reusken, Paul Schwering · 2021

In recent literature several derivations of incompressible Navier-Stokes type equations that model the dynamics of an evolving fluidic surface have been presented. These derivations differ in the physical principles used in the modeling ap…

Frontmatter Open

Maxim A. Olshanskii, Andrea Bonito, Markus Bachmayr, Michele Benzi, Susanne C. Brenner , et al. · 2021

featuring contemporary research in all areas of Numerical Mathematics.This includes the development, analysis and implementation of new and innovative methods in Numerical Linear Algebra,

Optimal preconditioners for a Nitsche stabilized fictitious domain\n finite element method Open

Sven Groß, Arnold Reusken · 2021

In this paper we consider a class of fictitious domain finite element methods\nknown from the literature. These methods use standard finite element spaces on\na fixed unfitted triangulation combined with the Nitsche technique and a ghost\n…

Optimal preconditioners for a Nitsche stabilized fictitious domain finite element method Open

Sven Groß, Arnold Reusken · 2021

In this paper we consider a class of fictitious domain finite element methods known from the literature. These methods use standard finite element spaces on a fixed unfitted triangulation combined with the Nitsche technique and a ghost pen…

Finite element discretization methods for velocity-pressure and stream function formulations of surface Stokes equations Open

Philip Brandner, Thomas Jankuhn, Simon Praetorius, Arnold Reusken, Axel Voigt · 2021

In this paper we study parametric TraceFEM and parametric SurfaceFEM (SFEM) discretizations of a surface Stokes problem. These methods are applied both to the Stokes problem in velocity-pressure formulation and in stream function formulati…

Analysis of the Schwarz Domain Decomposition Method for the Conductor-like Screening Continuum Model Open

Arnold Reusken, Benjamin Stamm · 2021

We study the Schwarz overlapping domain decomposition method applied to the\nPoisson problem on a special family of domains, which by construction consist\nof a union of a large number of fixed-size subdomains. These domains are\nmotivated…

Analysis of finite element methods for surface vector-Laplace eigenproblems Open

Arnold Reusken · 2020

In this paper we study finite element discretizations of a surface vector-Laplace eigenproblem. We consider two known classes of finite element methods, namely one based on a vector analogon of the Dziuk-Elliott surface finite element meth…

Finite element error analysis of surface Stokes equations in stream function formulation Open

Philip Brandner, Arnold Reusken · 2020

We consider a surface Stokes problem in stream function formulation on a simply connected oriented surface Γ ⊂ ℝ 3 without boundary. This formulation leads to a coupled system of two second order scalar surface partial differential equatio…

Inf-sup stability of the trace $\mathbf {P}_2$–$P_1$ Taylor–Hood elements for surface PDEs Open

Maxim A. Olshanskii, Arnold Reusken, Alexander Zhiliakov · 2020

The paper studies a geometrically unfitted finite element method (FEM), known\nas trace FEM or cut FEM, for the numerical solution of the Stokes system posed\non a closed smooth surface. A trace FEM based on standard Taylor-Hood\n(continuo…

Error analysis of higher order trace finite element methods for the surface Stokes equations Open

Thomas Jankuhn, Maxim A. Olshanskii, Arnold Reusken, Alexander Zhiliakov · 2020

The paper studies a higher order unfitted finite element method for the Stokes system posed on a surface in three-dimensional space. The method employs generalized Taylor-Hood finite element pairs on tetrahedral bulk mesh to discretize the…

A Multigrid Method for Unfitted Finite Element Discretizations of Elliptic Interface Problems Open

Thomas Ludescher, Sven Groß, Arnold Reusken · 2020

We consider discrete Poisson interface problems resulting from linear unfitted finite elements, also called cut finite elements (CutFEM). Three of these unfitted finite element methods known from the literature are studied. All three metho…

Multilevel preconditioning of stabilized unfitted finite element discretizations Open

Thomas Ludescher, Christoph Lehrenfeld, Arnold Reusken · 2020

In this thesis new methods for the iterative solution of interface problems are presented. Two problem classes are considered: the Poisson interface problem and the Stokes interface problem. These are model problems for mass transport and …

Trace finite element methods for surface vector-Laplace equations Open

Thomas Jankuhn, Arnold Reusken · 2019

In this paper we analyze a class of trace finite element methods for the discretization of vector-Laplace equations. A key issue in the finite element discretization of such problems is the treatment of the constraint that the unknown vect…

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