Ragnar Winther

• Falk, Richard & Winther, Ragnar (2025). Local Space-Preserving Decompositions for the Bubble Transform. Foundations of Computational Mathematics. ISSN 1615-3375. doi: 10.1007/s10208-025-09700-2. Full text in Research Archive Show summary

  The bubble transform is a procedure to decompose differential forms, which are piecewise smooth with respect to a given triangulation of the domain, into a sum of local bubbles. In this paper, an improved version of a construction in the setting of the de Rham complex previously proposed by the authors is presented. The major improvement in the decomposition is that unlike the previous results, in which the individual bubbles were rational functions with the property that groups of local bubbles summed up to preserve piecewise smoothness, the new decomposition is strictly space-preserving in the sense that each local bubble preserves piecewise smoothness. An important property of the transform is that the construction only depends on the given triangulation of the domain and is independent of any finite element space. On the other hand, all the standard piecewise polynomial spaces are invariant under the transform. Other key properties of the transform are that it commutes with the exterior derivative, is bounded in L2, and satisfies the stable decomposition property.

• Falk, Richard S. & Winther, Ragnar (2023). CONSTRUCTION OF POLYNOMIAL PRESERVING COCHAIN EXTENSIONS BY BLENDING. Mathematics of Computation. ISSN 0025-5718. 92(342), p. 1575–1594. doi: 10.1090/mcom/3819. Full text in Research Archive Show summary

  A classical technique to construct polynomial preserving extensions of scalar functions defined on the boundary of an 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 which map traces of k-forms defined on the boundary of the simplex to k-forms defined in the interior. Furthermore, the extensions are cochain maps, i.e., they commute with the exterior derivative.

• Tai, Xue-Cheng; Winther, Ragnar; Zhang, Xiaodi & Zheng, Weiying (2023). A uniform preconditioned for a Newton algorithm for total-variation minimisation and minimum-surface problems. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 61(5), p. 2062–2083. doi: 10.1137/22M1512776. Full text in Research Archive Show summary

  Solution methods for the nonlinear PDE 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 iterative methods have been proposed, such as Chan–Golub–Mulet method. In this paper, we propose a new Newton–Krylov solver for primal-dual finite element discretization of the ROF model and the minimum-surface model. The method is so simple that we just need to use some diagonal preconditioners during the iterations. Theoretically, the proposed preconditioners are further proved to be robust and optimal with respect to the mesh size, the penalization parameter, the regularization parameter, and the iterative step; essentially, it is a parameter-independent preconditioner. We first discretize the primal-dual system by using mixed finite element methods and then linearize the discrete system by Newton’s method. Exploiting the well-posedness of the linearized problem on appropriate Sobolev spaces equipped with proper norms, we propose block diagonal preconditioners for the corresponding system solved with the minimum residual method. Numerical results are presented to support the theoretical results.

• Falk, Richard S. & Winther, Ragnar (2022). The Bubble Transform and the de Rham Complex. Foundations of Computational Mathematics. ISSN 1615-3375. doi: 10.1007/s10208-022-09589-1. Full text in Research Archive Show summary

  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. The present paper contains a similar study for differential forms. From a simplicial mesh T of the domain Ω, we build a map which decomposes piecewise smooth k-forms into a sum of local bubbles supported on appropriate macroelements. The key properties of the decomposition are that it commutes with the exterior derivative and preserves the piecewise polynomial structure of the standard finite element spaces of k-forms. Furthermore, the transform is bounded in L2 and also on the appropriate subspace consisting of k-forms with exterior derivatives in L2.

• Hu, Kaibo & Winther, Ragnar (2021). Well conditioned frames for high order finite element methods. Journal of Computational Mathematics. ISSN 0254-9409. 39(3), p. 333–357. doi: 10.4208/JCM.2001-M2018-0078. Full text in Research Archive Show summary

  The purpose of this paper is to discuss representations of high order C0 finite element spaces on simplicial meshes in any dimension. When computing with high order piecewise polynomials the conditioning of the basis is likely to be important. The main result of this paper is a construction of representations by frames such that the associated L2 condition number is bounded independently of the polynomial degree. To our knowledge, such a representation has not been presented earlier. The main tools we will use for the construction is the bubble transform, introduced previously in [1], and properties of Jacobi polynomials on simplexes in higher dimensions. We also include a brief discussion of preconditioned iterative methods for the finite element systems in the setting of representations by frames.

• Lee, Jeonghun & Winther, Ragnar (2018). Local coderivatives and approximation of Hodge Laplace problems. Mathematics of Computation. ISSN 0025-5718. 87(314), p. 2709–2735. doi: 10.1090/mcom/3315. Full text in Research Archive Show summary

  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 property is an inherent consequence of the mixed formulation of these methods, and can be argued to be an undesired effect of these schemes. As a consequence, it has been argued, at least in special settings, that more local methods may have improved properties. In the present paper, we construct such methods by relying on a careful balance between the choice of finite element spaces, degrees of freedom, and numerical integration rules. Furthermore, we establish key convergence estimates based on a standard approach of variational crimes.

• Bærland, Trygve; Lee, Jeonghun J.; Mardal, Kent-Andre & Winther, Ragnar (2017). Weakly Imposed Symmetry and Robust Preconditioners for Biot’s Consolidation Model. Computational Methods in Applied Mathematics. ISSN 1609-4840. 17(3), p. 377–396. doi: 10.1515/cmam-2017-0016. Full text in Research Archive Show summary

  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 applications in science, medicine, and engineering. A challenge in many of these applications is that the model parameters range over several orders of magnitude. Therefore, discretization procedures which are well behaved with respect to such variations are needed. The focus of the present paper will be on the construction of preconditioners, such that the preconditioned discrete systems are well-conditioned with respect to variations of the model parameters as well as refinements of the discretization. As a byproduct, we also obtain preconditioners for linear elasticity that are robust in the incompressible limit.

• Lee, Jeonghun; Mardal, Kent-Andre & Winther, Ragnar (2017). Parameter-robust discretization and preconditioning of Biot's consolidation model. SIAM Journal on Scientific Computing. ISSN 1064-8275. 39(1), p. A1–A24. doi: 10.1137/15M1029473. Full text in Research Archive Show summary

  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 respect to these variations. The purpose of this paper is to study finite element discretizations of this model and construct block diagonal preconditioners for the discrete Biot systems. The approach taken here is to consider the stability of the problem in nonstandard or weighted Hilbert spaces and employ the operator preconditioning approach. We derive preconditioners that are robust with respect to both the variations of the parameters and the mesh refinement. The parameters of interest are small time-step sizes, large bulk and shear moduli, and small hydraulic conductivity. © 2017, Society for Industrial and Applied Mathematics.

• Tveito, Aslak; Langtangen, Hans; Winther, Ragnar & Engquist, Björn (2015). Computational Partial Differential Equations. In Engquist, Björn (Eds.), Encyclopedia of Applied and Computational Mathematics. Springer. ISSN 9783540705284. p. 271–278. Full text in Research Archive
• Falk, Richard S. & Winther, Ragnar (2015). Double complexes and local cochain projections. Numerical Methods for Partial Differential Equations. ISSN 0749-159X. 31(2), p. 541–551. doi: 10.1002/num.21922. Full text in Research Archive Show summary

  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 unbounded canonical projections defined by the degrees of freedom. However, an undesired property of these bounded projections is that, in contrast to the canonical projections, they are nonlocal. The purpose of this article is to discuss a recent alternative construction of bounded cochain projections, which also are local. A key tool for the new construction is the structure of a double complex, resembling the Čech-de Rham double complex of algebraic topology.

• Falk, Richard S. & Winther, Ragnar (2015). The Bubble Transform: A New Tool for Analysis of Finite Element Methods. Foundations of Computational Mathematics. ISSN 1615-3375. 16(1), p. 297–328. doi: 10.1007/s10208-015-9252-1. Full text in Research Archive Show summary

  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 Ω⊂Rn into a collection of functions with local support. In fact, for a given simplicial triangulation T of Ω, the associated bubble transform BT produces a decomposition of functions on Ω into a sum of functions with support on the corresponding macroelements. The transform is bounded in both L2 and the Sobolev space H1, it is local, and it preserves the corresponding continuous piecewise polynomial spaces. As a consequence, this transform is a useful tool for constructing local projection operators into finite element spaces such that the appropriate operator norms are bounded independently of polynomial degree. The transform is basically constructed by two families of operators, local averaging operators and rational trace preserving cutoff operators.

• Falk, Richard S. & Winther, Ragnar (2014). Local bounded cochain projections. Mathematics of Computation. ISSN 0025-5718. 83(290), p. 2631–2656. doi: 10.1090/S0025-5718-2014-02827-5. Full text in Research Archive Show summary

  First published in Mathematics of Computation Mathematics of Computation. 2014, 83 (290), published by the American Mathematical Society.

• Arnold, Douglas N.; Awanou, Gerard & Winther, Ragnar (2014). Nonconforming tetrahedral mixed finite elements for elasticity. Mathematical Models and Methods in Applied Sciences. ISSN 0218-2025. 24(4), p. 783–796. doi: 10.1142/S021820251350067X. Full text in Research Archive Show summary

  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 convergent for both the stress and displacement, and which is significantly simpler than any stable conforming mixed finite element method. The method may be viewed as the three-dimensional analogue of a previously developed element in two dimensions. As in that case, a variant of the method is proposed as well, in which the displacement approximation is reduced to piecewise rigid motions and the stress space is reduced accordingly, but the linear convergence is retained. Electronic version of an article published as Mathematical Models and Methods in Applied Sciences. 2014, 24 (4), 783-796 © World Scientific Publishing Company

• Christiansen, Snorre H & Winther, Ragnar (2013). On variational eigenvalue approximation of semidefinite operators. IMA Journal of Numerical Analysis. ISSN 0272-4979. 33(1), p. 164–189. doi: 10.1093/imanum/drs002. Full text in Research Archive
• Mardal, Kent-Andre; Schoberl, Joachim & Winther, Ragnar (2013). A uniformly stable Fortin operator for the Taylor-Hood element. Numerische Mathematik. ISSN 0029-599X. 123(3), p. 537–551. doi: 10.1007/s00211-012-0492-6. Full text in Research Archive
• Winther, Ragnar & Mardal, Kent-Andre (2011). Construction of preconditioners by mapping properties for systems of partial differential equations. In Axelsson, Owe & Karatson, Janos (Ed.), Efficient Preconditioned Solution Methods for Elliptic Partial Differential Equations. Bentham eBooks. ISSN 9781608052912. p. 65–83. Full text in Research Archive
• Mardal, Kent-Andre & Winther, Ragnar (2011). Preconditioning discretizations of systems of partial differential equations. Numerical Linear Algebra with Applications. ISSN 1070-5325. 18(1), p. 1–40. doi: 10.1002/nla.716. Full text in Research Archive
• Rognes, Marie & Winther, Ragnar (2010). Mixed Finite Element Methods for Viscoelasticity With Weak Symmetry. Mathematical Models and Methods in Applied Sciences (M3AS). 20, p. 955–985. doi: 10.1142/S0218202510004490. Full text in Research Archive Show summary

  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 derived and numerical experiments presented. The approach can be applied to general models for linear viscoelasticity and thus offers a unifed framework.

• Winther, Ragnar & Rognes, Marie Elisabeth (2010). Mixed finite element methods for linear viscoelasticity using weak symmetry. Mathematical Models and Methods in Applied Sciences. ISSN 0218-2025. 20(6), p. 955–985. doi: 10.1142/S0218202510004490. Full text in Research Archive
• Arnold, DN; Falk, RS & Winther, Ragnar (2010). FINITE ELEMENT EXTERIOR CALCULUS FROM HODGE THEORY TO NUMERICAL STABILITY. Bulletin of the American Mathematical Society. ISSN 0273-0979. 47(2), p. 281–354. Full text in Research Archive
• Karper, Trygve; Mardal, Kent-Andre & Winther, Ragnar (2009). Unified Finite Element Discretizations of Coupled Darcy-Stokes Flow. Numerical Methods for Partial Differential Equations. ISSN 0749-159X. 25(2), p. 311–326. Full text in Research Archive
• Arnold, Douglas N.; Falk, Richard S. & Winther, Ragnar (2009). Geometric decompositions and local bases for spaces of finite element differential forms. Computer Methods in Applied Mechanics and Engineering. ISSN 0045-7825. 198(21-26), p. 1660–1672. doi: 10.1016/j.cma.2008.12.017. Full text in Research Archive
• Hu, Qiya; Tai, Xue-Cheng & Winther, Ragnar (2009). A saddle point approach to the computation of harmonic maps. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 47(2), p. 1500–1523. doi: 10.1137/060675575. Full text in Research Archive
• Karper, Trygve Klovning; Mardal, Kent-Andre & Winther, Ragnar (2009). Unified Finite Element Discretizations of Coupled Darcy-Stokes Flow. Numerical Methods for Partial Differential Equations. ISSN 0749-159X. 25(2), p. 311–326. doi: 10.1002/num.20349. Full text in Research Archive
• Arnold, DN; Awanou, G & Winther, Ragnar (2008). Finite elements for symmetric tensors in three dimensions. Mathematics of Computation. ISSN 0025-5718. 77, p. 1229–1251. Full text in Research Archive
• Karper, Trygve Klovning; Mardal, Kent-Andre & Winther, Ragnar (2008). Unified finite element discretizations of coupled Darcy-Stokes flow. Numerical Methods for Partial Differential Equations. ISSN 0749-159X. 25(2), p. 311–326. Full text in Research Archive
• Christiansen, Snorre H & Winther, Ragnar (2008). Smoothed projections in finite element exterior calculus. Mathematics of Computation. ISSN 0025-5718. 77. Full text in Research Archive
• Arnold, DN; Falk, RS & Winther, Ragnar (2007). Mixed finite element methods for linear elasticity with weakly imposed symmetry. Mathematics of Computation. ISSN 0025-5718. 76, p. 1699–1723. Full text in Research Archive
• Mardal, Kent-Andre & Winther, Ragnar (2006). An Observation on Korn's Inequality for Nonconforming Finite Element Methods. Math. Comp.. 75, p. 1–6. Full text in Research Archive
• Tai, Xue-Cheng & Winther, Ragnar (2006). A discrete de Rham complex with enhanced smoothness. Calcolo. ISSN 0008-0624. 43(4), p. 287–306. Full text in Research Archive
• Klausen, Runhild Aae & Winther, Ragnar (2006). Convergence of multipoint flux approximations on quadrilateral grids. Numerical Methods for Partial Differential Equations. ISSN 0749-159X. 22, p. 1438–1454. Full text in Research Archive
• Klausen, Runhild Aae & Winther, Ragnar (2006). Robust convergence of multi point flux approximation on rough grids. Numerische Mathematik. ISSN 0029-599X. 104, p. 317–337. Full text in Research Archive
• Winther, Ragnar & Mardal, Kent-Andre (2006). An observation on Korn's inequality for nonconforming finite element methods. Mathematics of Computation. ISSN 0025-5718. 75(253), p. 1–6. Full text in Research Archive
• Christiansen, Snorre Harald & Winther, Ragnar (2006). On constraint preservation in numerical simulations of Yang-Mills equations. SIAM Journal on Scientific Computing. ISSN 1064-8275. 28. Full text in Research Archive
• Winther, Ragnar; Arnold, Douglas N. & Falk, Richard S. (2006). Finite element exterior calculus. Acta Numerica. ISSN 0962-4929. 15, p. 1–155. Full text in Research Archive
• Winther, Ragnar; Arnold, Douglas N. & Falk, Richard S. (2006). Differential complexes and stability of finite element methods. II: The elasticity complex. IMA Volumes in Mathematics and its Applications. ISSN 0940-6573. 142, p. 47–68. Full text in Research Archive
• Winther, Ragnar; Arnold, Douglas N. & Falk, Richard S. (2006). Differential complexes and stability of finite element methods. I: The de Rham complex. IMA Volumes in Mathematics and its Applications. ISSN 0940-6573. 142, p. 23–46. Full text in Research Archive
• Mardal, Kent-Andre & Winther, Ragnar (2004). Uniform Preconditioners for the Time Dependent Stokes Problem. Numerische Mathematik. ISSN 0029-599X. 98, p. 305–327. Full text in Research Archive
• Østby, Ivar & Winther, Ragnar (2004). Stability of a model of human granulopoiesis using continuous maturation. Journal of Mathematical Biology. ISSN 0303-6812. 49, p. 501–536. Full text in Research Archive
• Winther, Ragnar & Mardal, Kent-Andre (2004). Uniform preconditioners for the time dependent Stokes problem. Numerische Mathematik. ISSN 0029-599X. 98, p. 305–327. Full text in Research Archive
• Winther, Ragnar (2003). Mixed finite elements for elasticity in the stress-displacement formulation. Contemporary Mathematics - American Mathematical Society. ISSN 0271-4132. 329, p. 33–42. Full text in Research Archive
• Winther, Ragnar (2003). Nonconforming mixed methods for elasticity. Mathematical Models and Methods in Applied Sciences. ISSN 0218-2025. 13, p. 295–307. Full text in Research Archive
• Mardal, Kent-Andre; Tai, Xue-Cheng & Winther, Ragnar (2002). A Robust Finite Element Method for Darcy-Stokes Flow. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 40, p. 1605–1631. Full text in Research Archive
• Lyche, Tom Johan; Nilssen, Trygve Kastberg & Winther, Ragnar (2002). Preconditioned Iterative Methods for Scattered Data Interpolation. Advances in Computational Mathematics. ISSN 1019-7168. 17, p. 237–256. Full text in Research Archive
• Langtangen, Hans Petter; Mardal, Kent-Andre & Winther, Ragnar (2002). Numerical methods for incompressible viscous flow. Advances in Water Resources. ISSN 0309-1708. 25, p. 1125–1146. Full text in Research Archive
• Arnold, Douglas N. & Winther, Ragnar (2002). Mixed finite elements for elasticity. Numerische Mathematik. ISSN 0029-599X. 92, p. 401–419. Full text in Research Archive
• Mardal, Kent-Andre; Tai, Xue-Cheng & Winther, Ragnar (2002). A robust finite element method for Darcy-Stokes flow. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 40, p. 1605–1631. Full text in Research Archive

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• Tveito, Aslak & Winther, Ragnar (2009). Introduction to Partial Differential Equations; a Computational Approach. Springer. ISBN 9783540887041. Full text in Research Archive Show summary

  This is the softcover reprint of a popular book teaching the basic analytical and computational methods of partial differential equations. Standard topics such as separation of variables, Fourier analysis, maximum principles, and energy estimates are included. Prerequisites for this text are the very basics of calculus, linear algebra and ordinary differential equations. Numerical methods are included in the book to show the significance of computations in partial differential equations, and to illustrate the strong interaction between mathematical theory and numerical methods. Great care has been taken throughout the book to seek a sound balance between the analytical and numerical techniques. The authors present the material at an easy pace with exercises and projects ranging from the straightforward to the challenging. The text would be suitable for advanced undergraduate and graduate courses in mathematics and engineering, and it develops basic tools of computational science.

• Tveito, Aslak & Winther, Ragnar (2009). Introduction to Partial Differential Equations; A Computational Approach. Springer. ISBN 9783540887041. 392 p. Full text in Research Archive
• Tveito, Aslak & Winther, Ragnar (2002). Einführung in Die Theorie Der Partiellen Differentialgleichungen; Ein Numerischer Zugang. Springer. Full text in Research Archive
• Tveito, Aslak & Winther, Ragnar (2002). Einfuhrung in partielle differentalgleichungen, Ein numerischer zugang. Springer. ISBN 9783540424048. 392 p. Full text in Research Archive
• Winther, Ragnar & Tveito, Aslak (2002). Einfuhrung in die Theorie der partiellen Differentialgleichungen;Ein numericher Zugang. Springer. -1 p. Full text in Research Archive

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• Hu, Kaibo & Winther, Ragnar (2017). Well-Conditioned Frames for Finite Element Methods. Full text in Research Archive
• Winther, Ragnar (2014). Local bounded cochain projections and the bubble transform. Full text in Research Archive
• Winther, Ragnar (2014). Local bounded cochain projections and the bubble transform. Full text in Research Archive
• Mardal, Kent-Andre; Schoberl, Joachim & Winther, Ragnar (2012). A Uniform Inf-Sup Condition With Applications to Preconditioning. Full text in Research Archive
• Winther, Ragnar (2012). Uniform preconditioners and the Taylor-Hood method. Full text in Research Archive
• Winther, Ragnar (2012). Five Lectures on Finite Element Exterior Calculus. Full text in Research Archive
• Berre, Inga; Dahle, Helge K.; Karlsen, Kenneth Hvistendahl; Nordbotten, Jan Martin; Tai, Xue-Cheng & Winther, Ragnar (2012). Magne Espedal Memorial Issue. International Journal of Numerical Analysis & Modeling. ISSN 1705-5105. 9(3), p. I–II. Full text in Research Archive
• Winther, Ragnar (2011). Finite element exterior calculus - a link between topology and numerical analysis. Full text in Research Archive
• Winther, Ragnar (2011). Bounded cochain projections, why and how. Full text in Research Archive
• Winther, Ragnar (2011). Bounded cochain projections and finite elementexterior calculus. Full text in Research Archive
• Winther, Ragnar (2010). Introduction to finite element exterior calculus. Full text in Research Archive
• Winther, Ragnar (2010). Finite element exterior calculus - from Hodge theory to numerical stability. Full text in Research Archive
• Winther, Ragnar (2010). Cochain projections in finite element exterior calculus. Full text in Research Archive
• Winther, Ragnar (2009). Preconditioning discretizations of systems of partial differential equations. Full text in Research Archive
• Winther, Ragnar (2008). Mixed finite elements for linear elasticity and viscoelasticity. Full text in Research Archive
• Winther, Ragnar (2008). Finite element discretizations of the elasticity complex. Full text in Research Archive
• Winther, Ragnar (2008). Preconditioning saddle point problems arising from discretizations of partial differential equations (3 lectures). Full text in Research Archive
• Winther, Ragnar (2008). Finite element exterior calculus - a link between algebraic topology and numerical analysis. Full text in Research Archive
• Winther, Ragnar (2008). Smoothed projections in finite element exterior calculus. Full text in Research Archive
• Winther, Ragnar (2006). Finite element exterior calculus and applications. Full text in Research Archive
• Winther, Ragnar (2005). A discrete de Rham complex with enhanced smoothness. Full text in Research Archive
• Winther, Ragnar (2005). Mixed finite elements for elasticity, a constructive approach. Full text in Research Archive
• Winther, Ragnar (2005). Exact sequences and mixed finite elements for elasticity. Full text in Research Archive
• Winther, Ragnar (2004). Matematiske muligheter. Full text in Research Archive
• Winther, Ragnar (2004). Robust preconditioners for mixed systems. Full text in Research Archive
• Winther, Ragnar (2004). On the construction of mixed finite elements for elasticity. Full text in Research Archive
• Winther, Ragnar (2004). From the de Rham sequence to mixed finite elements for elasticity. Full text in Research Archive
• Winther, Ragnar (2004). Preconditioning saddle point systems. Full text in Research Archive
• Winther, Ragnar & Ellingsrud, Geir (2004). Matematiske muligheter. Full text in Research Archive
• Langtangen, Hans Petter; Mardal, Kent-Andre & Winther, Ragnar (2001). An efficient parallel iterative approach to the time-dependent Stokes problem. Full text in Research Archive
• Winther, Ragnar (2001). Robust finite elements for mixed systems. Full text in Research Archive
• Winther, Ragnar (2001). A robust finite element method for Darcy-Stokes flow. Full text in Research Archive
• Winther, Ragnar (2001). Robust preconditioners for mixed systems. Full text in Research Archive
• Muntingh, Agnar Georg Peder; Piene, Ragni & Winther, Ragnar (2011). Topics in Polynomial Interpolation Theory. Universitetet i Oslo. Full text in Research Archive Show summary

  This thesis studies two aspects of polynomial interpolation theory. The first part sets forth explicit formulas for the coefficients of polynomial interpolants to implicit functions. A formula for the higher-order derivatives of implicit functions appears as a limiting case of these formulas. The second part delves into certain configurations of points in space — generalized principal lattices — that are well suited as interpolation points. Applying the theory of real algebraic curves then allows the construction of many new examples of such configurations.

• Halvorsen, Tore Gunnar; Christiansen, Snorre H & Winther, Ragnar (2009). Structure preserving discretizations of wave equations from theoretical physics. Universitetet i Oslo. Full text in Research Archive
• Rognes, Marie Elisabeth & Winther, Ragnar (2008). MIXED FINITE ELEMENT METHODS FOR LINEAR VISCOELASTICITY USING WEAK SYMMETRY. Universitetet i Oslo. Full text in Research Archive Show summary

  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 derived and numerical experiments presented. The approach can be applied to general models for linear viscoelasticity and thus offers a unified framework.

• Christiansen, Snorre H & Winther, Ragnar (2006). Smoothed projections in finite element exterior calculus. Universitetet i Oslo. Full text in Research Archive Show summary

  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 effective tool in finite element exterior calculus. The advantage of these operators is that they are L2 bounded projections, and still they commute with the exterior derivative. In the present paper we generalize the construction of these smoothed projections, such that also non quasi--uniform meshes and essential boundary conditions are covered. The new tool introduced here is a space dependent smoothing operator which commutes with the exterior derivative.

• Klausen, Runhild Aae & Winther, Ragnar (2005). CONVERGENCE OF MULTI POINT FLUX APPROXIMATIONS ON QUADRILATERAL GRIDS. Universitetet i Oslo. Full text in Research Archive Show summary

  This paper presents a convergence analysis of the multi point 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 between the MPFA method and a mixed finite element method, using a specific numerical quadrature, such that the analysis of the MPFA method can be done in a finite element setting.

• Klausen, Runhild Aae & Winther, Ragnar (2005). ROBUST CONVERGENCE OF MULTI POINT FLUX APPROXIMATION ON ROUGH GRIDS. Universitetet i Oslo. Full text in Research Archive Show summary

  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 consider here a version where the flux approximation is derived directly in the physical space, and not on a reference cell. As a consequence, less regular grids are allowed. However, the extra cost is that the symmetry of the method is lost.

• Tai, Xue-Cheng & Winther, Ragnar (2005). A DISCRETE DE RHAM COMPLEX WITH ENHANCED SMOOTHNESS. Universitetet i Oslo. Full text in Research Archive Show summary

  Discrete de Rham complexes are fundamental tools in the construction of stable elements for mixed 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 some extra smoothness compared to the standard requirements. The motivation for this construction is to produce discretization which have uniform stability properties for certain families of singular perturbation problems. In particular, we will show how the spaces constructed here lead to discretizations of Stokes type systems which have uniform convergence properties as the Stokes flow approaches a Darcy flow.

• Christiansen, Snorre H & Winther, Ragnar (2004). On constraint preservation in numerical simulations of Yang-Mills equations. Universitetet i Oslo. Full text in Research Archive Show summary

  The problem of constraint preservation for discretizations of non-linear PDEs is addressed on 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 finite element spaces of Lie algebra valued differential forms. Numerical experiments indicate that simply restricting the variational formulation to the Galerkin spaces yields substantial drift in the charge, contrary to the linear Maxwell case. We then propose a fully discrete method constrained with Lagrange multipliers, for which we prove discrete charge conservation and observe excellent energy conservation.

• Winther, Ragnar & Mardal, Kent-Andre (2004). An observation on Korn's inequality for nonconforming finite element methods. Universitetet i Oslo. Full text in Research Archive
• Winther, Ragnar & Østby, Ivar (2003). Stability of a model of human granulopoiesis using continuous maturation. Universitetet i Oslo. Full text in Research Archive
• Arnold, Douglas N. & Winther, Ragnar (2002). Nonconforming mixed elements for elasticity. Full text in Research Archive
• Mardal, Kent-Andre & Winther, Ragnar (2002). Uniform preconditioners for the time dependent Stokes problem. Full text in Research Archive
• Arnold, Douglas N. & Winther, Ragnar (2002). Mixed finite elements for elasticity in the stress-displacement formulation. Full text in Research Archive
• Langtangen, Hans Petter; Mardal, Kent-Andre & Winther, Ragnar (2002). Numerical methods for incompressible viscous flow. Full text in Research Archive
• Arnold, Douglas N. & Winther, Ragnar (2001). Mixed finite elements for elasticity. Full text in Research Archive

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