arXiv subjects

Arnold Reusken

Publications and source records attributed to Arnold Reusken.

At least 37 records · Page 2Linked to original sources

arXiv2021

Optimal preconditioners for a Nitsche stabilized fictitious domain finite element method

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 penalty stabilization. As a model problem we consider the application of such a method to the Poisson equation. We introduce and analyze a new class of preconditioners that is based on a subspace decomposition approach. The finite element space is split into two subspaces, where one subspace is spanned by all nodal basis functions corresponding to nodes on the boundary of the fictitious domain and the other space is spanned by all remaining nodal basis functions. We will show that this splitting is stable, uniformly in the discretization parameter and in the location of the problem boundary in the triangulation. We also prove that the Galerkin discretization in the first subspace leads to a uniformly well-conditioned matrix and that the Galerkin discretization in the second subspace is uniformly equivalent to a standard finite element discretization of a Poisson equation on the fictitious domain with homogeneous Dirichlet boundary conditions. Results of numerical experiments that illustrate optimality of such a preconditioner are included.

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arXiv2021

Analysis of the Schwarz domain decomposition method for the conductor-like screening continuum model

We study the Schwarz overlapping domain decomposition method applied to the Poisson problem on a special family of domains, which by construction consist of a union of a large number of fixed-size subdomains. These domains are motivated by applications in computational chemistry where the subdomains consist of van der Waals balls. As is usual in the theory of domain decomposition methods, the rate of convergence of the Schwarz method is related to a stable subspace decomposition. We derive such a stable decomposition for this family of domains and analyze how the stability "constant" depends on relevant geometric properties of the domain. For this, we introduce new descriptors that are used to formalize the geometry for the family of domains. We show how, for an increasing number of subdomains, the rate of convergence of the Schwarz method depends on specific local geometry descriptors and on one global geometry descriptor. The analysis also naturally provides lower bounds in terms of the descriptors for the smallest eigenvalue of the Laplace eigenvalue problem for this family of domains.

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arXiv2020

Analysis of finite element methods for surface vector-Laplace eigenproblems

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 method and one based on the so-called trace finite element technique. A key ingredient in both classes of methods is a penalization method that is used to enforce tangentiality of the vector field in a weak sense. This penalization and the perturbations that arise from numerical approximation of the surface lead to essential nonconformities in the discretization of the variational formulation of the vector-Laplace eigenproblem. We present a general abstract framework applicable to such nonconforming discretizations of eigenproblems. Error bounds both for eigenvalue and eigenvector approximations are derived that depend on certain consistency and approximability parameters. Sharpness of these bounds is discussed. Results of a numerical experiment illustrate certain convergence properties of such finite element discretizations of the surface vector-Laplace eigenproblem.

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arXiv2020

Inf-sup stability of the trace P2-P1 Taylor-Hood elements for surface PDEs

The paper studies a geometrically unfitted finite element method (FEM), known as trace FEM or cut FEM, for the numerical solution of the Stokes system posed on a closed smooth surface. A trace FEM based on standard Taylor-Hood (continuous P2-P1) bulk elements is proposed. A so-called volume normal derivative stabilization, known from the literature on trace FEM, is an essential ingredient of this method. The key result proved in the paper is an inf-sup stability of the trace P2-P1 finite element pair, with the stability constant uniformly bounded with respect to the discretization parameter and the position of the surface in the bulk mesh. Optimal order convergence of a consistent variant of the finite element method follows from this new stability result and interpolation properties of the trace FEM. Properties of the method are illustrated with numerical examples.

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arXiv2020

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

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 Stokes system on embedded surface. Stability and optimal order convergence results are proved. The proofs include a complete quantification of geometric errors stemming from approximate parametric representation of the surface. Numerical experiments include formal convergence studies and an example of the Kelvin-Helmholtz instability problem on the unit sphere.

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arXiv2019

Finite Element Error Analysis of Surface Stokes Equations in Stream Function Formulation

We consider a surface Stokes problem in stream function formulation on a simply connected oriented surface $Γ\subset \mathbb{R}^3$ without boundary. This formulation leads to a coupled system of two second order scalar surface partial differential equations (for the stream function and an auxiliary variable). To this coupled system a trace finite element discretization method is applied. The main topic of the paper is an error analysis of this discretization method, resulting in optimal order discretization error bounds. The analysis applies to the surface finite element method of Dziuk-Elliott, too. We also investigate methods for reconstructing velocity and pressure from the stream function approximation. Results of numerical experiments are included.

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arXiv2019

Higher order Trace Finite Element Methods for the Surface Stokes Equation

In this paper a class of higher order finite element methods for the discretization of surface Stokes equations is studied. These methods are based on an unfitted finite element approach in which standard Taylor-Hood spaces on an underlying bulk mesh are used. For treating the constraint that the velocity must be tangential to the surface a penalty method is applied. Higher order geometry approximation is obtained by using a parametric trace finite element technique, known from the literature on trace finite element methods for scalar surface partial differential equations. Based on theoretical analyses for related problems, specific choices for the parameters in the method are proposed. Results of a systematic numerical study are included in which different variants are compared and convergence properties are illustrated.

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arXiv2019

Trace Finite Element Methods for Surface Vector-Laplace Equations

In this paper we analyze a class of trace finite element methods (TraceFEM) 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 vector field must be tangential to the surface (``tangent condition''). We study three different natural techniques for treating the tangent condition, namely a consistent penalty method, a simpler inconsistent penalty method and a Lagrange multiplier method. A main goal of the paper is to present an analysis that reveals important properties of these three different techniques for treating the tangent constraint. A detailed error analysis is presented that takes the approximation of both the geometry of the surface and the solution of the partial differential equation into account. Error bounds in the energy norm are derived that show how the discretization error depends on relevant parameters such as the degree of the polynomials used for the approximation of the solution, the degree of the polynomials used for the approximation of the level set function that characterizes the surface, the penalty parameter and the degree of the polynomials used for the approximation of Lagrange multiplier.

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arXiv2018

Incompressible fluid problems on embedded surfaces: Modeling and variational formulations

Governing equations of motion for a viscous incompressible material surface are derived from the balance laws of continuum mechanics. The surface is treated as a time-dependent smooth orientable manifold of codimension one in an ambient Euclidian space. We use elementary tangential calculus to derive the governing equations in terms of exterior differential operators in Cartesian coordinates. The resulting equations can be seen as the Navier-Stokes equations posed on an evolving manifold. We consider a splitting of the surface Navier-Stokes system into coupled equations for the tangential and normal motions of the material surface. We then restrict ourselves to the case of a geometrically stationary manifold of codimension one embedded in $\Bbb{R}^n$. For this case, we present new well-posedness results for the simplified surface fluid model consisting of the surface Stokes equations. Finally, we propose and analyze several alternative variational formulations for this surface Stokes problem, including constrained and penalized formulations, which are convenient for Galerkin discretization methods.

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arXiv2018

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

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 methods rely on Nitsche s method to incorporate the interface conditions. The main topic of the paper is the development of a multigrid method, based on a novel prolongation operator for the unfitted finite element space and an interface smoother that is designed to yield robustness for large jumps in the diffusion coefficients. Numerical results are presented which illustrate efficiency of this multigrid method and demonstrate its robustness properties with respect to variation of the mesh size, location of the interface and contrast in the diffusion coefficients.

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arXiv2018

A time dependent Stokes interface problem: well-posedness and space-time finite element discretization

In this paper a time dependent Stokes problem that is motivated by a standard sharp interface model for the fluid dynamics of two-phase flows is studied. This Stokes interface problem has discontinuous density and viscosity coefficients and a pressure solution that is discontinuous across an evolving interface. This strongly simplified two-phase Stokes equation is considered to be a good model problem for the development and analysis of finite element discretization methods for two-phase flow problems. In view of the unfitted finite element methods that are often used for two-phase flow simulations, we are particularly interested in a well-posed variational formulation of this Stokes interface problem in a Euclidean setting. Such well-posed weak formulations, which are not known in the literature, are the main results of this paper. Different variants are considered, namely one with suitable spaces of divergence free functions, a discrete-in-time version of it, and variants in which the divergence free constraint in the solution space is treated by a pressure Lagrange multiplier. The discrete-in-time variational formulation involving the pressure variable for the divergence free constraint is a natural starting point for a space-time finite element discretization. Such a method is introduced and results of numerical experiments with this method are presented.

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arXiv2018

Discontinuous Galerkin Time Discretization Methods for Parabolic Problems with Linear Constraints

We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that fit into the general framework are the heat equation with inhomogeneous (time dependent) Dirichlet boundary conditions and the time dependent Stokes equation with an inhomogeneous divergence constraint. Two common ways of treating such linear constraints, namely explicit or implicit (via Lagrange multipliers) are studied. These different treatments lead to different variational formulations of the parabolic problem. For these formulations we introduce a modification of the standard discontinuous Galerkin (DG) time discretization method in which an appropriate projection is used in the discretization of the constraint. For these discretizations (optimal) error bounds, including superconvergence results, are derived. Discretization error bounds for the Lagrange multiplier are presented. Results of experiments confirm the theoretically predicted optimal convergence rates and show that without the modification the (standard) DG method has sub-optimal convergence behavior.

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arXiv2018

A finite element method for the surface Stokes problem

We consider a Stokes problem posed on a 2D surface embedded in a 3D domain. The equations describe an equilibrium, area-preserving tangential flow of a viscous surface fluid and serve as a model problem in the dynamics of material interfaces. In this paper, we develop and analyze a Trace finite element method (TraceFEM) for such a surface Stokes problem. TraceFEM relies on finite element spaces defined on a fixed, surface-independent background mesh which consists of shape-regular tetrahedra. Thus, there is no need for surface parametrization or surface fitting with the mesh. The TraceFEM treated here is based on $P_1$ bulk finite elements for both the velocity and the pressure. In order to enforce the velocity vector field to be tangential to the surface we introduce a penalty term. The method is straightforward to implement and has an $O(h^2)$ geometric consistency error, which is of the same order as the approximation error due to the $P_1$--$P_1$ pair for velocity and pressure. We prove stability and optimal order discretization error bounds in the surface $H^1$ and $L^2$ norms. A series of numerical experiments is presented to illustrate certain features of the proposed TraceFEM.

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arXiv2017

A Trace Finite Element Method for Vector-Laplacians on Surfaces

We consider a vector-Laplace problem posed on a 2D surface embedded in a 3D domain, which results from the modeling of surface fluids based on exterior Cartesian differential operators. The main topic of this paper is the development and analysis of a finite element method for the discretization of this surface partial differential equation. We apply the trace finite element technique, in which finite element spaces on a background shape-regular tetrahedral mesh that is surface-independent are used for discretization. In order to satisfy the constraint that the solution vector field is tangential to the surface we introduce a Lagrange multiplier. We show well-posedness of the resulting saddle point formulation. A discrete variant of this formulation is introduced which contains suitable stabilization terms and is based on trace finite element spaces. For this method we derive optimal discretization error bounds. Furthermore algebraic properties of the resulting discrete saddle point problem are studied. In particular an optimal Schur complement preconditioner is proposed. Results of a numerical experiment are included.

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arXiv2017

$L^2$-error analysis of an isoparametric unfitted finite element method for elliptic interface problems

In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method was introduced which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. This method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. In the paper [C. Lehrenfeld, A. Reusken, \emph{Analysis of a High Order Finite Element Method for Elliptic Interface Problems}, arXiv 1602.02970, Accepted for publication in IMA J. Numer. Anal.] an a priori error analysis of the method applied to an interface problem is presented. The analysis reveals optimal order discretization error bounds in the $H^1$-norm. In this paper we extend this analysis and derive optimal $L^2$-error bounds.

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arXiv2017

Trace Finite Element Methods for PDEs on Surfaces

In this paper we consider a class of unfitted finite element methods for discretization of partial differential equations on surfaces. In this class of methods known as the Trace Finite Element Method (TraceFEM), restrictions or traces of background surface-independent finite element functions are used to approximate the solution of a PDE on a surface. We treat equations on steady and time-dependent (evolving) surfaces. Higher order TraceFEM is explained in detail. We review the error analysis and algebraic properties of the method. The paper navigates through the known variants of the TraceFEM and the literature on the subject.

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arXiv2017

Analysis of a high order unfitted finite element method for elliptic interface problems

In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. We consider a new unfitted finite element method which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. The method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. Both components, the piecewise planar interface reconstruction and the parametric mapping are easy to implement. In this paper we present an a priori error analysis of the method applied to an interface problem. The analysis reveals optimal order error bounds for the geometry approximation and for the finite element approximation, for arbitrary high order discretization. The theoretical results are confirmed in numerical experiments.

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arXiv2017

Analysis of a high order Trace Finite Element Method for PDEs on level set surfaces

We present a new high order finite element method for the discretization of partial differential equations on stationary smooth surfaces which are implicitly described as the zero level of a level set function. The discretization is based on a trace finite element technique. The higher discretization accuracy is obtained by using an isoparametric mapping of the volume mesh, based on the level set function, as introduced in [C. Lehrenfeld, \emph{High order unfitted finite element methods on level set domains using isoparametric mappings}, Comp. Meth. Appl. Mech. Engrg. 2016]. The resulting trace finite element method is easy to implement. We present an error analysis of this method and derive optimal order $H^1(Γ)$-norm error bounds. A second topic of this paper is a unified analysis of several stabilization methods for trace finite element methods. Only a stabilization method which is based on adding an anisotropic diffusion in the volume mesh is able to control the condition number of the stiffness matrix also for the case of higher order discretizations. Results of numerical experiments are included which confirm the theoretical findings on optimal order discretization errors and uniformly bounded condition numbers.

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