Snorre Harald Christiansen

• Christiansen, Snorre H (2024). On the definition of curvature in Regge calculus. IMA Journal of Numerical Analysis. ISSN 0272-4979. 44(5), p. 2698–2715. doi: 10.1093/imanum/drad095. Full text in Research Archive Show summary

  Regge calculus defines a curvature for piecewise constant metrics on simplicial complexes subject to a partial continuity requirement. We prove that if a part of the complex is embedded in a Euclidean space, by a piecewise affine map, and we perform smoothing by convolution there, then the smoothed metrics have a densitized scalar curvature that converges, in the sense of measures, to that defined by Regge, as the smoothing parameter goes to zero.

• Christiansen, Snorre H; Gopalakrishnan, Jay; Guzmán, Johnny & Hu, Kaibo (2023). A discrete elasticity complex on three-dimensional Alfeld splits. Numerische Mathematik. ISSN 0029-599X. 156, p. 159–204. doi: 10.1007/s00211-023-01381-9. Full text in Research Archive Show summary

  We construct conforming finite element elasticity complexes on the Alfeld splits of tetrahedra. The complex consists of vector fields and symmetric tensor fields, interlinked via the linearized deformation operator, the linearized curvature operator, and the divergence operator, respectively. The construction is based on an algebraic machinery that derives the elasticity complex from de Rham complexes, and smoother finite element differential forms.

• Christiansen, Snorre H; Halvorsen, Tore Gunnar & Scheid, Claire (2023). Convergence of a discretization of the Maxwell–Klein–Gordon equation based on finite element methods and lattice gauge theory. Numerical Methods for Partial Differential Equations. ISSN 0749-159X. 39, p. 3271–3308. doi: 10.1002/num.23008. Full text in Research Archive Show summary

  Abstract The Maxwell–Klein–Gordon equations are a set of coupled nonlinear time‐dependent wave equations, used to model the interaction of an electromagnetic field with a particle. The solutions, expressed with a magnetic vector potential, are invariant under gauge transformations. This characteristic implies a constraint on the solution fields that might be broken at the discrete level. In this article, we propose and study a constraint preserving numerical scheme for this set of equations in dimension 2. At the semidiscrete level, we combine conforming Finite Element discretizations with the so‐called Lattice Gauge Theory to design a compatible gauge invariant discretization, leading to preservation of a discrete constraint. Relying on energy techniques and compactness arguments, we establish the convergence of this semidiscrete scheme, without a priori knowledge of the solution. Finally, at the fully discrete level, we propose a compatible explicit time‐integration strategy of leapfrog type. We implement the resulting fully discrete scheme and provide assessment on academic scenarios.

• Christiansen, Snorre H & Hu, Kaibo (2022). Finite Element Systems for Vector Bundles: Elasticity and Curvature. Foundations of Computational Mathematics. ISSN 1615-3375. doi: 10.1007/s10208-022-09555-x. Full text in Research Archive Show summary

  Abstract We develop a theory of finite element systems, for the purpose of discretizing sections of vector bundles, in particular those arising in the theory of elasticity. In the presence of curvature, we prove a discrete Bianchi identity. In the flat case, we prove a de Rham theorem on cohomology groups. We check that some known mixed finite elements for the stress–displacement formulation of elasticity fit our framework. We also define, in dimension two, the first conforming finite element spaces of metrics with good linearized curvature, corresponding to strain tensors with Saint-Venant compatibility conditions. Cochains with coefficients in rigid motions are given a key role in relating continuous and discrete elasticity complexes.

• Erik, Burman; Christiansen, Snorre H & Hansbo, Peter (2020). Application of a minimal compatible element to incompressible and nearly incompressible continuum mechanics. Computer Methods in Applied Mechanics and Engineering. ISSN 0045-7825. 369. doi: 10.1016/j.cma.2020.113224. Full text in Research Archive Show summary

  In this note we will explore some applications of the recently constructed piecewise affine, H1-conforming element that fits in a discrete de Rham complex (Christiansen and Hu, 2018). In particular we show how the element leads to locking free methods for incompressible elasticity and viscosity robust methods for the Brinkman model.

• Christiansen, Snorre H & Licht, Martin Werner (2019). Poincaré-Friedrichs inequalities of complexes of discrete distributional differential forms. BIT Numerical Mathematics. ISSN 0006-3835. p. 1–27. doi: 10.1007/s10543-019-00784-1. Full text in Research Archive Show summary

  We derive bounds for the constants in Poincaré–Friedrichs inequalities with respect to mesh-dependent norms for complexes of discrete distributional differential forms. A key tool is a generalized flux reconstruction which is of independent interest. The results apply to piecewise polynomial de Rham sequences on bounded domains with mixed boundary conditions.

• Christiansen, Snorre H; Hu, Kaibo & Sande, Espen (2019). Poincare path integrals for elasticity. Journal des Mathématiques Pures et Appliquées. ISSN 0021-7824. doi: 10.1016/j.matpur.2019.06.002. Full text in Research Archive
• Christiansen, Snorre H & Hu, Kaibo (2018). Generalized finite element systems for smooth differential forms and Stokes' problem. Numerische Mathematik. ISSN 0029-599X. 140(2), p. 327–371. doi: 10.1007/s00211-018-0970-6. Full text in Research Archive Show summary

  We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously introduced notion of finite element systems, and the examples include conforming mixed finite elements for Stokes’ equation. In dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes. In dimension 3 we define one low order complex, which may be branched into Whitney forms at a chosen index. Stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension. The finite element spaces all consist of composite polynomials. The framework guarantees some nice properties of the spaces, in particular the existence of commuting interpolators. It also shows that some of the examples are minimal spaces.

• Christiansen, Snorre H; Hu, Jun & Hu, Kaibo (2017). Nodal finite element de Rham complexes. Numerische Mathematik. ISSN 0029-599X. Published ahead of print, p. 1–36. doi: 10.1007/s00211-017-0939-x. Full text in Research Archive Show summary

  We construct 2D and 3D finite element de Rham sequences of arbitrary polynomial degrees with extra smoothness. Some of these elements have nodal degrees of freedom and can be considered as generalisations of scalar Hermite and Lagrange elements. Using the nodal values, the number of global degrees of freedom is reduced compared with the classical Nédélec and Brezzi–Douglas–Marini finite elements, and the basis functions are more canonical and easier to construct. Our finite elements for H(div) with regularity r=2 coincide with the nonstandard elements given by Stenberg (Numer Math 115(1):131–139, 2010). We show how regularity decreases in the finite element complexes, so that they branch into known complexes. The standard de Rham complexes of Whitney forms and their higher order version can be regarded as the family with the lowest regularity. The construction of the new families is motivated by finite element systems. This is a post-peer-review, pre-copyedit version of an article published in Numerische Mathematik. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00211-017-0939-x

• Christiansen, Snorre H (2017). On eigenmode approximation for Dirac equations: differential forms and fractional Sobolev spaces. Mathematics of Computation. ISSN 0025-5718. 87(310), p. 547–580. doi: 10.1090/mcom/3233. Full text in Research Archive Show summary

  We comment on the discretization of the Dirac equation using finite element spaces of differential forms. In order to treat perturbations by low order terms, such as those arising from electromagnetic fields, we develop some abstract discretization theory and provide estimates in fractional order Sobolev spaces for finite element systems. Eigenmode convergence is proved, as well as optimal convergence orders, assuming a flat background metric on a periodic domain. This research was first published in Mathematics of Computation. © American Mathematical Society.

• Christiansen, Snorre H & Gillette, Andrew (2016). Constructions of some minimal finite element systems. Modélisation mathématique et analyse numérique. ISSN 0764-583X. 50(3), p. 833–850. doi: 10.1051/m2an/2015089. Full text in Research Archive Show summary

  Within the framework of finite element systems, we show how spaces of differential forms may be constructed, in such a way that they are equipped with commuting interpolators and contain prescribed functions, and are minimal under these constraints. We show how various known mixed finite element spaces fulfill such a design principle, including trimmed polynomial differential forms, serendipity elements and TNT elements. We also comment on virtual element methods and provide a dimension formula for minimal compatible finite element systems containing polynomials of a given degree on hypercubes. © EDP Sciences, SMAI 2016

• Christiansen, Snorre H & Rapetti, Francesca (2016). On high order finite element spaces of differential forms. Mathematics of Computation. ISSN 0025-5718. 85(298), p. 517–548. doi: 10.1090/mcom/2995. Full text in Research Archive Show summary

  We show how the high order finite element spaces of differential forms due to Raviart-Thomas-Nédelec-Hiptmair fit into the framework of finite element systems, in an elaboration of the finite element exterior calculus of Arnold-Falk-Winther. Based on observations by Bossavit, we provide new low order degrees of freedom. As an alternative to existing choices of bases, we provide canonical resolutions in terms of scalar polynomials and Whitney forms. First published in Mathematics of Computation in volume 85, 2016, published by the American Mathematical Society.

• 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
• Christiansen, Snorre H (2013). Upwinding in finite element systems of differential forms. London Mathematical Society Lecture Notes. ISSN 0076-0552. 403, p. 45–71. doi: 10.1017/CBO9781139095402.004. Full text in Research Archive Show summary

  We provide a notion of finite element system, that enables the construction spaces of differential forms, which can be used for the numerical solution of variationally posed partial difeerential equations. Within this framework, we introduce a form of upwinding, with the aim of stabilizing methods for the purposes of computational uid dynamics, in the vanishing viscosity regime. Published as the Smale Prize Lecture in: Foundations of computational mathematics, Budapest 2011, London Mathematical Society Lecture Note Series, 403, Cambridge University Press, 2013.

• Christiansen, Snorre H & Halvorsen, Tore Gunnar (2012). A simplicial gauge theory. Journal of Mathematical Physics. ISSN 0022-2488. 53(3). doi: 10.1063/1.3692167. Full text in Research Archive
• Christiansen, Snorre H & Halvorsen, Tore Gunnar (2011). A gauge invariant discretization on simplicial grids of the Schrödinger eigenvalue problem in an electromagnetic field. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 49(1), p. 331–345. doi: 10.1137/090757502. Full text in Research Archive
• Christiansen, Snorre H & Scheid, Claire (2011). Convergence of a constrained finite element discretization of the Maxwell Klein Gordon equation. Modélisation mathématique et analyse numérique. ISSN 0764-583X. 45(4), p. 739–760. doi: 10.1051/m2an/2010100. Full text in Research Archive
• Christiansen, Snorre H (2011). On the linearization of Regge calculus. Numerische Mathematik. ISSN 0029-599X. 119(4), p. 613–640. doi: 10.1007/s00211-011-0394-z. Full text in Research Archive
• Christiansen, Snorre H & Halvorsen, Tore Gunnar (2011). Discretizing the Maxwell-Klein-Gordon equation by the Lattice Gauge Theory formalism. IMA Journal of Numerical Analysis. ISSN 0272-4979. 31(1), p. 1–24. doi: 10.1093/imanum/drp019. Full text in Research Archive
• Christiansen, Snorre H; Munthe-Kaas, Hans & Owren, Brynjulf (2011). Topics in structure-preserving discretization. Acta Numerica. ISSN 0962-4929. 20, p. 1–119. doi: 10.1017/S096249291100002X. Full text in Research Archive
• Christiansen, Snorre H (2010). Minimal mixed finite elements on polyhedra. Comptes rendus. Mathematique. ISSN 1631-073X. 348(3-4), p. 217–221. doi: 10.1016/j.crma.2010.01.017. Full text in Research Archive
• Christiansen, Snorre H (2009). Foundations of Finite Element Methods for Wave Equations of Maxwell Type. In Soomere, Tarmo (Eds.), Applied wave mathematics. Springer. ISSN 9783642005848. p. 335–393. Full text in Research Archive
• Christiansen, Snorre H (2009). On the div-curl lemma in a Galerkin setting. Calcolo. ISSN 0008-0624. 46(3), p. 211–220. doi: 10.1007/s10092-009-0008-7. Full text in Research Archive
• Christiansen, Snorre H & Halvorsen, Tore Gunnar (2009). Convergence of lattice gauge theory for Maxwell's equations. BIT Numerical Mathematics. ISSN 0006-3835. 49(4), p. 645–667. doi: 10.1007/s10543-009-0242-z. Full text in Research Archive
• Andriulli, FP; Cools, K; Bagci, H; Olyslager, F; Buffa, A & Christiansen, Snorre H [Show all 7 contributors for this article] (2008). A multiplicative Calderon preconditioner for the electric field integral equation. IEEE Transactions on Antennas and Propagation. ISSN 0018-926X. 56, p. 2398–2412. doi: 10.1109/TAP.2008.926788. Full text in Research Archive
• Christiansen, Snorre H (2008). A construction of spaces of compatible differential forms on cellular complexes. Mathematical Models and Methods in Applied Sciences. ISSN 0218-2025. 18, p. 739–757. doi: 10.1142/S021820250800284X. Full text in Research Archive Show summary

  Given a cellular complex, we construct spaces of differential forms which form a complex under the exterior derivative, which is isomorphic to the cochain complex of the cellular complex. The construction applies in particular to subsets of Euclidean space divided into polyhedra, for which it provides, for each k, a space of k-forms with a basis indexed by the set of k-dimensional cells. In the framework of mimetic finite differences, the construction provides a conforming reconstruction operator. The construction requires auxiliary spaces of differential forms on each cell, for which we provide two examples. When the cells are simplexes, the construction can be used to recover the standard mixed finite element spaces also called Whitney forms. We can also recover the dual finite elements previously constructed by A. Buffa and the author on the barycentric refinement of a two-dimensional mesh.

• 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
• Buffa, Annalisa & Christiansen, Snorre H (2007). A dual finite element complex on the barycentric refinement. Mathematics of Computation. ISSN 0025-5718. 76, p. 1743–1769. Full text in Research Archive
• Christiansen, Snorre H (2007). Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension. Numerische Mathematik. ISSN 0029-599X. 107, p. 87–106. doi: 10.1007/s00211-007-0081-2. 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
• Buffa, Annalisa & Christiansen, Snorre Harald (2005). A dual finite element complex on the barycentric refinement. Comptes rendus. Mathematique. ISSN 1631-073X. 340. Full text in Research Archive
• Christiansen, Snorre Harald (2005). A div-curl lemma for edge elements. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 43, p. 116–126. Full text in Research Archive
• Christiansen, Snorre H (2004). A characterization of second order differential operators on finite element spaces. Mathematical Models and Methods in Applied Sciences. ISSN 0218-2025. 14(12), p. 1881–1892. Full text in Research Archive
• Christiansen, Snorre H (2004). Discrete Fredholm properties and convergence estimates for the electric field integral equation. Mathematics of Computation. ISSN 0025-5718. 73(245), p. 143–167. Full text in Research Archive

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• Christiansen, Snorre H (2016). On curvature of piecewise flat manifolds. Full text in Research Archive
• Christiansen, Snorre H (2016). Uniformly stable finite elements for convection diffusion equations. Full text in Research Archive
• Christiansen, Snorre H (2013). Systèmes d’éléments finis de formes différentielles, variantes sous le vent. Full text in Research Archive
• Christiansen, Snorre H (2013). Finite element systems of differential forms. Full text in Research Archive
• Christiansen, Snorre H (2013). Dual finite elements for Calderon preconditioning of the Electric Field Integral Equation. Full text in Research Archive
• Christiansen, Snorre H (2013). Finite element systems of differential forms and applications to upwinding. Full text in Research Archive
• Christiansen, Snorre H (2013). Constraint preservation in Lattice Gauge Theory. Full text in Research Archive
• Christiansen, Snorre H (2011). Giovani matematici crescono: Snorre Christiansen. [Internet]. http://maddmaths.simai.eu/giovani-matematici-crescono/snorre. Full text in Research Archive
• Christiansen, Snorre H (2011). Spectral correctness of linearized Regge calculus. Full text in Research Archive
• Christiansen, Snorre H (2011). Upwinding in finite element spaces of differential forms. Full text in Research Archive
• Christiansen, Snorre H (2010). A construction of minimal finite element spaces of differential forms on polyhedra. Full text in Research Archive
• Christiansen, Snorre H (2010). Introduction to mixed finite elements. Full text in Research Archive
• Christiansen, Snorre H (2010). Éléments finis mixtes minimaux sur les polyèdres. Full text in Research Archive
• Christiansen, Snorre H (2010). Coupling Maxwell with Schrödinger equations using Lattice Gauge Theory. Full text in Research Archive
• Christiansen, Snorre H (2010). Mixed finite elements as inverse systems of differential forms. Full text in Research Archive
• Christiansen, Snorre H (2010). Introduction to Regge calculus and its linearization. Full text in Research Archive
• Christiansen, Snorre H (2010). Éléments finis mixtes minimaux sur les polyèdres. Full text in Research Archive
• Christiansen, Snorre H (2010). Lattice gauge theory meets finite elements. Full text in Research Archive
• Christiansen, Snorre H (2010). Gauge invariant discretizations of Schrödinger equations in an electromagnetic field. Full text in Research Archive
• Christiansen, Snorre H (2010). Applications of lattice gauge theory to the coupling of Maxwell with Klein-Gordon and Schroedinger equations. Full text in Research Archive
• Christiansen, Snorre H (2010). On the linearization of Regge Calculus in 3D. Full text in Research Archive
• Christiansen, Snorre H (2009). A framework for the discretization of differential forms on cellular complexes. Full text in Research Archive
• Christiansen, Snorre H (2009). Discretizing the Yang-Mills-Higgs equation. Full text in Research Archive
• Christiansen, Snorre H (2009). A discussion of the div-curl lemma in a Galerkin setting. Full text in Research Archive
• Christiansen, Snorre H (2009). On the div-curl lemma in a Galerkin setting. Full text in Research Archive
• Christiansen, Snorre H (2009). Finite elements and differential forms. Full text in Research Archive
• Christiansen, Snorre H (2009). Finite element exterior calculus on cellular complexes. Full text in Research Archive
• Christiansen, Snorre H (2008). Regge calculus as a non-standard finite element method. Full text in Research Archive
• Christiansen, Snorre H (2008). A construction of spaces of compatible differential forms on cellular complexes. Full text in Research Archive
• Christiansen, Snorre H (2008). A construction of spaces of mixed finite elements on cellular complexes. Full text in Research Archive
• Christiansen, Snorre H (2008). Discrete differential forms: from algebraic topology to approximation of Maxwell's equations. Full text in Research Archive
• Christiansen, Snorre H (2008). On the linearization of Regge calculus. Full text in Research Archive
• Christiansen, Snorre H (2007). Discrete differential forms: from algebraic topology to approximation of Maxwell's equations. Full text in Research Archive
• Christiansen, Snorre H (2007). Constraint preserving discretizations for some gauge invariant wave equations. Full text in Research Archive
• Christiansen, Snorre H (2007). A construction of mixed finite element spaces on polyhedral meshes. Full text in Research Archive
• Christiansen, Snorre H (2007). Constraint preserving schemes for some gauge invariant wave equations. Full text in Research Archive
• Christiansen, Snorre H (2006). A construction of mixed finite element spaces on polyhedral meshes. Full text in Research Archive
• Christiansen, Snorre H (2006). Skolen ble en døråpner. [Journal]. Dagbladet. Full text in Research Archive
• Christiansen, Snorre H (2006). Some homological algebra techniques in numerical analysis. Full text in Research Archive
• Christiansen, Snorre H (2006). On constraint preservation in numerical simulations of Yang-Mills equations. Full text in Research Archive
• Christiansen, Snorre H (2005). On constraint preservation in finite element discretizations of Yang-Mills equations. Full text in Research Archive
• Christiansen, Snorre H (2005). On divergence control in numerical simulations of Yang-Mills equations. Full text in Research Archive
• Christiansen, Snorre H (2005). On constraint preservation in numerical simulations of Yang-Mills equations. Full text in Research Archive
• Christiansen, Snorre H (2005). On numerical simulation of Yang-Mills equations. Full text in Research Archive
• Christiansen, Snorre H & Buffa, Annalisa (2005). A dual finite element complex on the barycentric refinement. Full text in Research Archive
• Christiansen, Snorre H (2005). On constraint preservation in numerical simulations of Yang-Mills equations. Full text in Research Archive
• Christiansen, Snorre H (2004). Div-curl lemma for edge elements. Full text in Research Archive
• Christiansen, Snorre H (2004). On divergence control in numerical simulations of Yang-Mills equations. Full text in Research Archive
• Christiansen, Snorre H (2004). A compensated compactness property of some finite element spaces. Full text in Research Archive
• Christiansen, Snorre H & Halvorsen, Tore Gunnar (2010). A simplicial gauge theory. http://arxiv.org/abs/1006.2059v3. Full text in Research Archive
• Christiansen, Snorre H (2010). On variational eigenvalue approximation of semidefinite operators. ArXiv. Full text in Research Archive
• Christiansen, Snorre H & Halvorsen, Tore Gunnar (2009). A gauge invariant discretization on simplicial grids of the Schrödinger eigenvalue problem in an electromagnetic field. Universitetet i Oslo. Full text in Research Archive Show summary

  We propose a method to compute approximate eigenpairs of the Schrödinger operator on a bounded domain in the presence of an electromagnetic field. The method is formulated for the simplicial grids that satisfy the discrete maximum principle. It combines techniques from lattice gauge theory and finite element methods, retaining the discrete gauge invariance of the former but allowing for non-congruent space elements as in the latter. The error in the method is studied in the framework of Strang's variational crimes, comparing with a standard Galerkin approach. For a smooth electromagnetic field the crime is of order the mesh width h, for a Coulomb potential it is of order h|log h|, and for a general finite energy electromagnetic field it is of order h1/2.

• Christiansen, Snorre H (2009). On the div-curl lemma in a Galerkin setting. Universitetet i Oslo. Full text in Research Archive
• Christiansen, Snorre H (2009). Foundations of finite element methods for wave equations of Maxwell type. Universitetet i Oslo. Full text in Research Archive Show summary

  The first part of the paper is an overview of the theory of approximation of wave equations by Galerkin methods. It treats convergence theory for linear second order evolution equations and includes studies of consistency and eigenvalue approximation. We emphasize differential operators, such as the curl, which have large kernels and use L2 stable interpolators preserving them. The second part is devoted to a framework for the construction of finite element spaces of differential forms on cellular complexes. Material on homological and tensor algebra as well as differential and discrete geometry is included. Whitney forms, their duals, their high order versions, their tensor products and their hp-versions all fit.

• Christiansen, Snorre H & Scheid, Claire (2009). Convergence of a constrained finite element discretization of the Maxwell Klein Gordon equation. Universitetet i Oslo. Full text in Research Archive Show summary

  As an example of a simple constrained geometric non-linear wave equation, we study a numerical approximation of the Maxwell Klein Gordon equation. We consider an existing constraint preserving semi-discrete scheme based on finite elements and prove its convergence in space dimension 2 for initial data of finite energy.

• Christiansen, Snorre H (2009). Éléments finis mixtes minimaux sur les polyèdres. Universitetet i Oslo. Full text in Research Archive Show summary

  Étant donné un complexe cellulaire constitué de polyèdres, plongé dans un espace Euclidien, nous construisons des espaces d'éléments finis de formes différentielles contenant celles qui sont polynomiales de degré maximal donné, ayant localement la propriété de suite exacte et d'extension, de telle sorte que parmi tous les espaces ayant ces trois propriétés, ils ont la plus petite dimension. Plus généralement nous construisons, pour tout système d'éléments finis inclus dans un système d'éléments finis compatible, un système d'éléments finis compatible intermédiaire et de dimension minimale.

• 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
• Christiansen, Snorre H & Halvorsen, Tore Gunnar (2008). Solving the Maxwell-Klein-Gordon equation in the Lattice Gauge Theory formalism. Universitetet i Oslo. Full text in Research Archive Show summary

  In this article we study the discretization of the Maxwell-Klein-Gordon equation from a variational point of view. We first solve the problem with an action corresponding to the Yee scheme for the Maxwell part, which is automatically gauge invariant, and a gauge invariant action for the Klein-Gordon part given by the Lattice Gauge Theory. The action is showed to be consistent with the continuous formulation, and the equations to be solved are derived from a discrete stationary action principle. Due to the gauge invariance, the local electric charge can be shown to be conserved through Noether's theorem. As this is an essential feature of the continuous model, this conservation can be viewed as the great achievement of this scheme. Thereafter we compare the above described scheme with a scheme that uses a standard finite difference approximation of the derivatives, and where the coupling between the scalar field and the gauge field is done in the simplest way. This scheme will posess a global gauge symmetry which ensures the conservation of global charge as in the hybrid case, but the scheme has no local symmetry and no locally conserved charge. At last we present some numerical results in the temporal gauge, shedding light on the theoretical discussion.

• Christiansen, Snorre H & Halvorsen, Tore Gunnar (2008). Convergence of lattice gauge theory for Maxwell's equations. Universitetet i Oslo. Full text in Research Archive Show summary

  In this article we show convergence of Lattice Gauge Theory with gauge group U(1) in the energy norm. This is done by stability analysis and comparison with the classical Yee-scheme which is convergent.

• Christiansen, Snorre H (2008). On the linearization of Regge calculus. Universitetet i Oslo. Full text in Research Archive Show summary

  We study the linearization of three dimensional Regge calculus around Euclidean metric. We provide an explicit formula for the corresponding quadratic form and relate it to the curl T curl operator which appears in the quadratic part of the Einstein-Hilbert action and also in the linear elasticity complex. We insert Regge metrics in a discrete version of this complex, equipped with densily defined and commuting interpolators. We show that the eigenpairs of the curl T curl operator, approximated using the quadratic part of the Regge action on Regge metrics, converge to their continuous counterparts, interpreting the computation as a non-conforming finite element method.

• Christiansen, Snorre H (2007). Constraint preserving schemes for some gauge invariant wave equations. Universitetet i Oslo. Full text in Research Archive Show summary

  We study discretizations of the Maxwell-Klein-Gordon equation as an example of a constrained geometric non-linear evolution partial differential equation. For the temporal gauge we propose a fully discrete scheme which preserves the non-linear constraint thanks to a special application of Lagrange multipliers. We show that the method generalizes to Hamiltonian wave equations whose kinetic and potential energy are both invariant under a group of transformations, even though the Galerkin spaces are not invariant. We then extend the method to the Lorenz gauge. Numerical results illustrate the discussion.

• 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.

• Christiansen, Snorre H (2005). Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension. Universitetet i Oslo. Full text in Research Archive Show summary

  We elaborate on the interpretation of some mixed finite element spaces in terms of differential forms. First we develop a framework in which we show how tools from algebraic topology can be applied to the study of their cohomological properties. The analysis applies in particular to certain hp finite element spaces, extending results in trivial topology often referred to as the exact sequence property. Then we define regularization operators. Combined with the standard interpolators they enable us to prove discrete Poincaré-Friedrichs inequalities and discrete Rellich compactness for finite element spaces of differential forms of arbitrary degree on compact manifolds of arbitrary dimension.

• Christiansen, Snorre H (2005). A construction of spaces of differential forms on cellular complexes. Institut Mittag-Leffler. Full text in Research Archive
• Buffa, Annalisa & Christiansen, Snorre H (2005). A dual finite element complex on the barycentric refinement. IMATI-CNR. Full text in Research Archive
• 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.

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