Ragnar Winther

University of Oslo

The purpose of this paper is to discuss a generalization of the bubble transform to differential forms. The bubble transform was discussed in Falk and Winther (Found Comput Math 16(1):297–328, 2016) for scalar valued functions, or zero-forms, and represents a new tool for the understanding of finite element spaces of arbitrary polynomial degree. Th...

Solution methods for the nonlinear partial differential equation of the Rudin-Osher-Fatemi (ROF) and minimum-surface models are fundamental for many modern applications. Many efficient algorithms have been proposed. First order methods are common. They are popular due to their simplicity and easy implementation. Some second order Newton-type iterat...

A classical technique to construct polynomial preserving extensions of scalar functions defined on the boundary of an $n$ simplex to the interior is to use so-called rational blending functions. The purpose of this paper is to generalize the construction by blending to the de Rham complex. More precisely, we define polynomial preserving extensions...

The purpose of this paper is to discuss a generalization of the bubble transform to differential forms. The bubble transform was discussed in a previous paper by the authors for scalar valued functions, or zero-forms, and represents a new tool for the understanding of finite element spaces of arbitrary polynomial degree. The present paper contains...

The standard mixed finite element approximations of Hodge Laplace problems associated with the de Rham complex are based on proper discrete subcomplexes. As a consequence, the exterior derivatives, which are local operators , are computed exactly. However, the approximations of the associated coderivatives are nonlocal. In fact, this nonlocal prope...

The purpose of this paper is to discuss representations of high order $C^0$ finite element spaces on simplicial meshes in any dimension. For high order methods the conditioning of the basis is likely to be important. However, so far there seems to be no generally accepted concept of "a well-conditioned basis", or a general strategy for how to obtai...

We discuss the construction of robust preconditioners for finite element approximations of Biot's consolidation model in poroelasticity. More precisely, we study finite element methods based on generalizations of the Hellinger-Reissner principle of linear elasticity, where the stress tensor is one of the unknowns. The Biot model has a number of app...

Biot's consolidation model in poroelasticity has a number of applications in science, medicine, and engineering. The model depends on various parameters, and in practical applications these parameters range over several orders of magnitude. A current challenge is to design discretization techniques and solution algorithms that are well-behaved with...

The construction of projection operators, which commute with the exterior derivative and at the same time are bounded in the proper Sobolev spaces, represents a key tool in the recent stability analysis of finite element exterior calculus. These so-called bounded cochain projections have been constructed by combining a smoothing operator and the un...

The purpose of this paper is to discuss the construction of a linear operator, referred to as the bubble transform, which maps scalar functions defined on a bounded domain $\Omega$ in $\mathbb{R}^n$ into a collection of functions with local support. In fact, for a given simplicial triangulation of $\Omega$, the associated bubble transform produces...

We construct a new Fortin operator for the lowest order Taylor–Hood element, which is uniformly stable both in $L^2$ and $H^1$ . The construction, which is restricted to two space dimensions, is based on a tight connection between a subspace of the Taylor–Hood velocity space and the lowest order Nedelec edge element. General shape regular trian...

We construct projections from the space of differential k-forms which belong to L2 and whose exterior derivative also belongs to L2, to finite dimensional subspaces of piecewise polynomial differential forms defined on a simplicial mesh. These projections have the properties that they commute with the exterior derivative and are bounded independent...

This paper presents a nonconforming finite element approximation of the space of symmetric tensors with square integrable divergence, on tetrahedral meshes. Used for stress approximation together with the full space of piecewise linear vector fields for displacement, this gives a stable mixed finite element method which is shown to be linearly conv...

A uniform inf-sup condition related to a parameter dependent Stokes problem is established. Such conditions are intimately connected to the construction of uniform preconditioners for the problem, i.e., preconditioners which behave uniformly well with respect to variations in the model parameter as well as the discretization parameter. For the pres...

The purpose of this chapter is to discuss a general approach to the construction of preconditioners for the linear systems of algebraic equations arising from discretizations of systems of partial differential equations. The discussion here is closely tied to our earlier paper [1], where we gave a comprehensive review of a mathematical theory for c...

This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for Partial Differential Equations in the summer of 2008. The main focus will be on an abstract approach to the construction of preconditioners for symmetric linear systems in a Hilbert space setti...

Small deformations of a viscoelastic body are considered through the linear Maxwell and Kelvin-Voigt models in the quasi-static equilibrium. A robust mixed finite element method, enforcing the symmetry of the stress tensor weakly, is proposed for these equations on simplicial tessellations in two and three dimensions. A priori error estimates are d...

Eigenvalue problems for semidefinite operators with infinite-dimensional kernels appear, for instance, in electromagnetics. Variational discretizations with edge elements have long been analysed in terms of a discrete compactness property. As an alternative, we show here how the abstract theory can be developed in terms of a geometric property call...

This introduction provides an overview of the missions and activities of the two involved research centres, CENS (Centre for Nonlinear Studies) in Tallinn, Estonia, and CMA (Centre of Mathematics for Applications) in Oslo, Norway. It also gives a description of the main features of the EU FP6 Marie Curie Transfer of Knowledge project CENS-CMA, from...

This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theo...

In this article, we discuss some new finite element methods for flows which are governed by the linear stationary Stokes system on one part of the domain and by a second order elliptic equation derived from Darcy's law in the rest of the domain, and where the solutions in the two domains are coupled by proper interface conditions. All the methods p...

In this paper we consider numerical approximations of a constraint minimization problem, where the object function is a quadratic Dirichlet functional for vector fields and the interior constraint is given by a convex function. The solutions of this problem are usually referred to as harmonic maps. The solution is characterized by a nonlinear saddl...

We construct finite element subspaces of the space of symmetric tensors with square-integrable divergence on a three-dimensional domain. These spaces can be used to approximate the stress field in the classical Hellinger--Reissner mixed formulation of the elasticty equations, when standard discontinous finite element spaces are used to approximate...

We study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions. These spaces are generalizations of the classical finite element spaces for vector fields, frequently referred to as Raviart-Thomas, Brezzi-Douglas-Marini, and Nedelec spaces. In the present paper, we...

The development of smoothed projections, constructed by combining the canonical interpolation operators defined from the degrees of freedom with a smoothing operator, have proved to be an eective tool in finite element exterior calculus. The advan- tage of these operators is that they are L2 bounded projections, and still they commute with the exte...

A differential form is a field which assigns to each point of a domain an alternating multilinear form on its tangent space. The exterior derivative operation, which maps differential forms to differential forms of the next higher order, unifies the basic first order differential operators of calculus, and is a building block for a great variety of...

A close connection between the ordinary de Rham complex and a corresponding elasticity complex is utilized to derive new mixed finite element methods for linear elasticity. For a formulation with weakly imposed symmetry, this approach leads to methods which are simpler than those previously obtained. For example, we construct stable discretizations...

A close connection between the ordinary de Rham complex and a corresponding elasticity complex is utilized to derive new mixed finite element methods for linear elasticity. For a formulation with weakly imposed symmetry, this approach leads to methods which are simpler than those previously obtained. For example, we construct stable discretizations...

In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger--Reissner variational principle that only weakly imposes the symmetry condition...

This article presents a convergence analysis of the multipoint flux approximation control volume method, MPFA, in two space dimensions. The MPFA version discussed here is the so-called O-method on general quadrilateral grids. The discretization is based on local mappings onto a reference square. The key ingredient in the analysis is an equivalence...

This paper establishes the convergence of a multi point flux approximation control volume method on rough quadrilateral grids. By rough grids we refer to a family of refined quadrilateral grids where the cells are not required to approach parallelograms in the asymptotic limit. In contrast to previous convergence results for these methods we consid...

Finite element exterior calculus is an approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretizations which are compatible with the geometr...

Discrete de Rham complexes are fundamental tools in the construction of stable elements for some finite element methods. The purpose of this paper is to discuss a new discrete de Rham complex in three space dimensions, where the finite element spaces have extra smoothness compared to the standard requirements. The motivation for this construction i...

The problem of constraint preservation for discretizations of nonlinear PDEs is addressed in the example of the hyperbolic Yang-Mills equations in temporal gauge. These equations preserve a nonlinear divergence field analogous to the electric charge for Maxwell's equations. We introduce and discuss several discretizations of these equations on fini...

By utilizing a simple observation on traces of rigid motions we are able to strengthen a result of Brenner (2004) on Korn's inequality for nonconforming finite element methods. The approach here is tightly connected to the theory developed in Brenner's work. Our motivation for the analysis was the desire to show that a robust Darcy-Stokes element s...

A class of mathematical models involving a convection-reaction partial differential equation (PDE) is introduced with reference to recovering human granulopoiesis after high dose chemotherapy with stem cell support. The stability properties of the model are addressed by means of numerical investigations and analysis. A simplified model with prolife...

Implicit time stepping procedures for the time dependent Stokes problem lead to stationary singular perturbation problems at each time step. These singular perturbation problems are systems of saddle point type, which formally approach a mixed formulation of the Poisson equation as the time step tends to zero. Preconditioners for discrete analogous...

We present a family of pairs of finite element spaces for the unaltered Hellinger--Reissner variational principle using polynomial shape functions on a single triangular mesh for stress and displacement. There is a member of the family for each polynomial degree, beginning with degree two for the stress and degree one for the displacement, and each...

The scattered data interpolation problem in two space dimensions is formulated as a partial dierential equation with interpolating side conditions. The system is discretized by the Morley nite element space. The focus of this paper is to study preconditioned iterative methods for the corresponding discrete systems. We in- troduce block diagonal pre...

We construct first order, stable, nonconforming mixed finite elements for plane elasticity and analyze their convergence. The mixed method is based on the Hellinger-Reissner variational formulation in which the stress and displacement fields are the primary unknowns. The stress elements use polynomial shape functions but do not involve vertex degre...

We present an overview of the most common numerical solution strategies for the incompressible Navier–Stokes equations, including fully implicit formulations, artificial compressibility methods, penalty formulations, and operator splitting methods (pressure/velocity correction, projection methods). A unified framework that explains popular operator...

In diesem Kapitel gehen wir kurz auf die Fouriertransformation und ihrer Einsatzmöglichkeit bei der Lösung von Differentialgleichungen ein, deren räumlicher Definitionsbereich ganz ℝ ist.

In den letzten beiden Kapiteln haben wir uns mit Fourierreihen befasst. Unsere Untersuchungen wurden hauptsächlich durch die Rolle der Fourierreihen bei der Herleitung formaler Lösungen diverser partieller Differentialgleichungen wie der Wärmeleitungsgleichung, der Wellengleichung und der Poisson-Gleichung motiviert.

Im letzten Abschnitt haben wir eine sehr effektive Methode zur analytischen Lösung partieller Differentialgleichungen hergeleitet. Mit einfachen Techniken konnten wir eine explizite Formel zur Lösung vieler parabolischer Differentialgleichungen angeben. Die Untersuchung dieser analytischen Lösungen lehrt einiges über das qualitative Verhalten solch...

Mit der Lektüre dieses Buches beginnen Sie eine Reise durch den Dschungel der partiellen Differentialgleichungen. Wie in jedem Urwald gibt es hier einerseits ringsherum interessante Sehenswürdigkeiten andererseits aber auch einige gefährliche Flecken. Auf ihrer Reise durch diesen Dschungel benötigen Sie geeignete Orientierungshilfen und Werkzeuge,...

Im Kapitel 1 tauchten die Wellengleichung in Abschnitt 1.4.3 und die Wärmeleitungsgleichung in Abschnitt 1.4.4 auf. In den Anwendungen begegnet man diesen Gleichungen recht häufig. Sie werden daher oft als fundamentale Gleichungen bezeichnet. Wir werden uns in späteren Kapiteln erneut mit diesen Problemen befassen. Eine weitere fundamentale Gleichu...

Im vorliegenden Kapitel befassen wir uns mit Maximumprinzipien. Diese ermöglichen gewisse Aussagen über die Lösungen von Gleichungen ohne die Gleichungen lösen zu müssen.

Reaktions-Diffusionsgleichungen tauchen als mathematische Modelle einer Reihe wichtiger Anwendungsgebiete auf. So modellieren sie z.B. supraleitende Flüssigkeiten, Flammenausbreitungen, chemische Kinetiken, biochemische Reaktionen, Räuber-Beute-Modelle der Ökologie und so weiter. Sowohl numerische als auch analytische Untersuchungen von Reaktions-D...

Sei f eine auf dem Intervall [−1,1] stückweise stetige Funktion mit der Fourierreihe \fraca0 2 + åk = 1¥ (ak cos(kpx) + bk sin(kpx)). \frac{{\alpha _0 }} {2} + \sum\limits_{k = 1}^\infty {(\alpha _k \cos (k\pi x) + b_k \sin (k\pi x)).}

In den letzten Kapiteln waren die Fourierreihen das wichtigste Hilfsmittel bei der Herleitung formaler Lösungen partieller Differentialgleichungen. In diesem sowie den beiden folgenden Kapiteln werden wir uns ausführlicher mit Fourierreihen und formalen Lösungen beschäftigen. Die in den bisherigen Kapiteln aufgetretenen Fourierreihen können als Bei...

Die Poisson-Gleichung ist eine der fundamentalen partiellen Differentialgleichungen. Sie spielt in vielen Bereichen der mathematischen Physik, zum Beispiel bei der Behandlung von Flußproblemen sowie bei Aufgaben der Elektrostatik eine wichtige Rolle. Die Poisson-Gleichung tauchte bereits während unserer Untersuchungen von Maximumprinzipien für harm...

Die Entwicklungen der mathematischen Physik, der Analysis und der Lösungsmethoden für partielle Differentialgleichungen sind sehr eng miteinander verflochten. Es ist oft schwer, klare Grenzen zwischen diesen Gebieten zu ziehen. Ganz besonders trifft dies auf die Fourieranalyse zu. Der französische Physiker Joseph Fourier (1768–1830) begründete dies...

In diesem Kapitel widmen wir uns den Anfangs-Randwertaufgaben für die Wellengleichung in einer Raumdimension. Insbesondere werden wir durch Trennung der Variablen formale Lösungen herleiten, die Eindeutigkeit einer Lösung mit Hilfe der Energiemethode beweisen und Eigenschaften von Lösungsapproximationen untersuchen, die sich aus finite Differenzenm...

Finite element methods for some elliptic fourth order singular perturbation problems are discussed. We show that if such problems are discretized by the nonconforming Morley method, in a regime close to second order elliptic equations, then the error deteriorates. In fact, a counterexample is given to show that the Morley method diverges for the re...

There have been many e#orts, dating back four decades, to develop stable mixed finite elements for the stress-displacement formulation of the plane elasticity system. This requires the development of a compatible pair of finite element spaces, one to discretize the space of symmetric tensors in which the stress field is sought, and one to discretiz...

Finite element methods for a family of systems of singular perturbation problems of a saddle point structure are discussed. The system is approximately a linear Stokes problem when the perturbation parameter is large, while it degenerates to a mixed formulation of Poisson's equation as the perturbation parameter tends to zero. It is established, ba...

The general theme of this project is to study numerical methods for systems of partial diffential equations which depend on one or more critical parameters. Typically, we are interested in systems which change type as a critical perturbation parameter tend to zero. Our goal is to construct numerical methods with convergence properties which are uni...

Summary. We consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the Hilbert spaces and in three dimensions. We show that if appropriate finite element spaces and appropriate additive or multiplicative Schwarz smoothers are used, then the multigrid V-cycle...

We consider iterative methods for the solution of the linear system of equations arising from the mixed finite element discretization of the Reissner--Mindlin plate model. We show how to construct a symmetric positive definite block diagonal preconditioner such that the resulting linear system has spectral condition number independent of both the m...

Finite element approximations for the Dirichlet problem associated a second-order elliptic differential equation are studied. The purpose of this paper is to discuss domain embedding preconditioners for the discrete systems. The essential boundary condition on the interior interface is removed by introducing Lagrange multipliers. The associated dis...

We consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the Hilbert spaces H(div) and H(curl) in three dimensions. We show that if appropriate finite element spaces and appropriate additive or multiplicative Schwarz smoothers are used, then the multigrid V...

this paper is to summarize the work of [AFW97]. We consider iterative methods for the solution of indefinite linear systems of equations arising from discretizations of the Reissner--Mindlin plate model. Like the biharmonic plate model, the Reissner--Mindlin model is a two-dimensional plate model which approximates the behavior of a thin linearly e...

this paper we study block diagonal preconditioners for mixed systems derived from the Dirichlet problems for second order elliptic equations. The main purpose is to discuss how an embedding of the original computational domain into a simpler extended domain can be utilized in this case. We show that a family of uniform preconditioners for the corre...

We consider a simple model of the motions of a viscoelastic solid. The model consists of a two by two system of conservation laws including a strong relaxation term. We establish the existence of a BV-solution of this system for any positive value of the relaxation parameter. We also show that this solution is stable with respect to the perturbatio...

In an earlier paper we constructed and analyzed a multigrid preconditioner for the system of linear algebraic equations arising from the finite element discretization of boundary value problems associated to the differential operator I - grad div. In this paper we analyze the procedure without assuming that the underlying domain is convex and show...

Some recent results on the rate of convergence towards equilibrium for some 2 × 2 systems of conservation laws which include stiff relaxation terms are presented. We focus our attention on certain specific model problems. We will discuss the possibility of deriving properties like bounds on the total variation, stability with respect the initial da...

In Chapter 1 above we encountered the wave equation in Section 1.4.3 and the heat equation in Section 1.4.4. These equations occur rather frequently in applications, and are therefore often referred to as fundamental equations. We will return to these equations in later chapters. Another fundamental equation is Poisson’s equation, given by $$ - \su...

The purpose of this chapter is to study initial-boundary value problems for the wave equation in one space dimension. In particular, we will derive formal solutions by a separation of variables technique, establish uniqueness of the solution by energy arguments, and study properties of finite difference approximations.

In the previous chapter we derived a very powerful analytical method for solving partial differential equations. By using straightforward techniques, we were able find an explicit formula for the solution of many partial differential equations of parabolic type. By studying these analytical solutions, we can learn a lot about the qualitative behavi...

You are embarking on a journey in a jungle called Partial Differential Equations. Like any other jungle, it is a wonderful place with interesting sights all around, but there are also certain dangerous spots. On your journey, you will need some guidelines and tools, which we will start developing in this introductory chapter.

In the previous chapters Fourier series have been the main tool for obtaining formal solutions of partial differential equations. The purpose of the present chapter and the two following chapters is to give a more thorough analysis of Fourier series and formal solutions. The Fourier series we have encountered in earlier chapters can be thought of a...

The purpose of this chapter is to study maximum principles. Such principles state something about the solution of an equation without having to solve it.

The two previous chapters have been devoted to Fourier series. Of course, the main motivation for the study of Fourier series was their appearance in formal analytic solutions of various partial differential equations like the heat equation, the wave equation, and Poisson’s equation.

Let f be a piecewise continuous function defined on [-1, 1] with a full Fourier series given by $$ \frac{{a_0 }} {2} + \sum\limits_{k = 1}^\infty {\left( {a_k \cos \left( {k\pi x} \right) + b_k \sin \left( {k\pi x} \right)} \right).} $$ . converge to the function f ?” If we here refer to convergence in the mean square sense, then a partial answer t...

In this chapter, we briefly discuss the Fourier transform and show how this transformation can be used to solve differential equations where the spatial domain is all of ℝ.

Poisson’s equation is a fundamental partial differential equation which arises in many areas of mathematical physics, for example in fluid flow, flow in porous media, and electrostatics. We have already encountered this equation in Section 6.4 above, where we studied the maximum principle for harmonic functions. As a corollary of the maximum princi...

We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I − g r a d ⁡ div - \operatorname {\mathbf {grad}}\operatorname {div} . The natural setting for such problems is in the Hilbert space H ( div ) (\operatorname {di...

A straightforward semi-implicit finite-difference method approximating a system of conservation laws including a stiff relaxation term is analyzed. We show that the error, measured in L-1, is bounded by O(root Delta t) independent of the stiffness, where the time step Delta t represents the mesh size. As a simple corollary we obtain that solutions...

. In an earlier paper we constructed and analyzed a multigrid preconditioner for the system of linear algebraic equations arising from the finite element discretization of boundary value problems associated to the differential operator I Gamma grad div. In this paper we analyze the procedure without assuming that the underlying domain is convex and...

We consider iterative methods for the solution of the linear system of equations arising from the mixed finite element discretization of the Reissner--Mindlin plate model. We show how to construct a symmetric positive definite block diagonal preconditioner such that the resulting linear system has spectral condition number independent of both the m...

We sudy domain embedding preconditioners for discrete linear systems approximating the Dirichlet problem associated with a second-order elliptic equation. We observe that if a mixed finite element discretization is used, then such a preconditioner can be constructed in a straightforward manner from the H(div)-inner product. We also use the H(div)-i...

We analyze a simple system of conservation laws with a strong relaxation term. Wellposedness of the Cauchy problem, in the framework of BV-solutions, is proved. Furthermore, we prove that the solutions converge towards the solution of an equilibrium model as the relaxation time ffi ? 0 tends to zero. Finally, we show that the difference between an...

We consider the solution of the system of linear algebraic equa- tions which arises from the,nite element discretization of boundary,value problems,associated to the dierential operator I grad div. The natural setting for such problems is in the Hilbert space H (div) and the variational formulation is based on the inner product in H (div). We show...

In this paper we discuss how certain saddle point problems, arising from discretizations of partial differential equations, should be preconditioned in order to obtain iterative methods which converge with a rate independent of the discretization parameters. The results for the discrete systems are motivated from corresponding results for the conti...

We analyze a system of conservation laws in two space dimensions with a stiff relaxation term. A semi-implicit finite difference method approximating the system is studied and an error bound of order\(\mathcal{O}(\sqrt {\Delta t} )\) measured inL 1 is derived. This error bound is independent of the relaxation time δ > 0. Furthermore, it is proved t...

. Explicit and semi--implicit finite difference schemes approximating nonhomogenous scalar conservation laws are analyzed. Optimal error bounds independent of the stiffness of the underlying equation are presented. Key words. Hyperbolic conservation law, source term, finite difference scheme, error estimate. AMS(MOS) subject classifications. 35L65,...

The order of convergence for operator splitting applied to conservation laws with source terms is studied. The operator splitting procedure is based on local solutions of the associated homogeneous conservation law and an ordinary differential equation. We prove that, for scalar problems, the error introduced by the splitting is linear with respect...

It is established that an interior penalty method applied to secondorder elliptic problems gives rise to a local operator which is spectrally equivalent to the corresponding nonlocal operator arising from the mixed finite element method. This relation can be utilized in order to construct preconditioners for the discrete mixed system. As an example...

Explicit and semi-implicit finite-difference schemes approximating nonhomogeneous scalar conservation laws are analyzed. Optimal error bounds independent of the stiffness of the underlying equation are presented.

Some simple examples of nonstrictly hyperbolic conservation laws are presented that admit certain instability properties with respect to $L^1 $-perturbations of the initial data. One of the examples shows that a piecewise constant approximation generates an approximate solution with erroneous qualitative behaviour; another example shows that the ap...

A rigorous proof of an error estimate for a finite difference scheme applied to a 2×2 nonstrictly hyperbolic system of conservation laws is presented. It is established that the error in the L1-norm is of order (Δx)1/2, where Δx denotes the mesh size. This convergence estimate is optimal since, for special data, the system degenerates to a scalar e...

Domain decomposition preconditioners for the linear systems arising from mixed finite element discretizations of second-order elliptic boundary value problems are proposed. The preconditioners are based on subproblems with either Neumann or Dirichlet boundary conditions on the interior boundary. The preconditioned systems have the same structure as...

A standard upwind scheme is used to solve non-strictly hyperbolic systems of conservation laws arising in enhanced oil recovery. The convergence rate is estimated either theoretically or from numerical experiments.