Jon Eivind Vatne studied mathematics at the University of Bergen, specializing in algebraic geometry. After finishing his doctorate in 2002, he had temporary positions as Post.Doc. and associate professor at the University of Bergen (funded by the Norwegian Research Council) and NTNU. He then held a permanent position at Western Norway University of Applied Sciences until he joined BI full-time from August 2021.
Publications
Boon, Wietse; Holmen, Daniel Førland, Nordbotten, Jan Martin & Vatne, Jon Eivind (2025)
The Hodge-Laplacian on the Čech-de Rham complex governs coupled problems
In this paper we present a method for computing the dihedral angle sums (and their two-sided estimates) of cartesian and skew product polytopes provided the sums of dihedral angles (or their estimates) are known for the factors. In addition, a formula for computing the number of facets of such product polytopes is derived. The method proposed is very universal and illustrated by several examples. The estimates
Korotov, Sergey & Vatne, Jon Eivind (2024)
Conforming simplicial partitions of product-decomposed polytopes
We propose some approaches for the generation of conforming simplicial partitions with various regularity properties for polytopic domains that are products or a union of products, thus generalizing our earlier results. The techniques presented can be used for finite element simulations of higher-dimensional problems.
Various angle characteristics are used (e.g. in finite element methods or computer graphics) when evaluating the quality of computational meshes which may consist, in the three-dimensional case, of tetrahedra, prisms, hexahedra and pyramids. Thus, it is of interest to derive (preferably tight) bounds for dihedral angle sums, i.e. sums of angles between faces, of such mesh elements. For tetrahedra this task was solved by Gaddum in 1952. For pyramids, this was resolved by Korotov, Lund and Vatne in 2022. In this paper, we compute tight bounds for the remaining two cases, hexahedra and prisms.
Korotov, Sergey; Lund, Lars Fredrik Kirkebø & Vatne, Jon Eivind (2022)
Tight bounds for the dihedral angle sums of a pyramid
We prove that eight dihedral angles in a pyramid with an arbitrary quadrilateral
base always sum up to a number in the interval (3π, 5π). Moreover, for any number in
(3π, 5π) there exists a pyramid whose dihedral angle sum is equal to this number, which
means that the lower and upper bounds are tight. Furthermore, the improved (and tight)
upper bound 4π is derived for the class of pyramids with parallelogramic bases. This includes
pyramids with rectangular bases, often used in finite element mesh generation and analysis.
Korotov, Sergey & Vatne, Jon Eivind (2021)
Preserved Structure Constants for Red Refinements of Product Elements
In this paper we discuss some strategy for red refinements of product elements and show that there are certain structure characteristics (d-sines of angles formed by certain edges in the initial partition) which remain constant during refinement processes. Such a property immediately implies the validity of the so-called maximum angle condition, which is a strongly desired property in interpolation theory and finite element analysis. Our construction also gives a clear refinement scheme preserving shape regularity.
Khademi, Ali; Korotov, Sergey & Vatne, Jon Eivind (2021)
On mesh regularity conditions for simplicial finite elements
We are interested in differential forms on mixed-dimensional geometries, in the sense of a domain containing sets of d-dimensional manifolds, structured hierarchically so that each d-dimensional manifold is contained in the boundary of one or more d+1-dimensional manifolds. On any given d-dimensional manifold, we then consider differential operators tangent to the manifold as well as discrete differential operators (jumps) normal to the manifold. The combined action of these operators leads to the notion of a semi-discrete differential operator coupling manifolds of different dimensions. We refer to the resulting systems of equations as mixed-dimensional, which have become a popular modeling technique for physical applications including fractured and composite materials. We establish analytical tools in the mixed-dimensional setting, including suitable inner products, differential and codifferential operators, Poincaré lemma, and Poincaré–Friedrichs inequality. The manuscript is concluded by defining the mixed-dimensional minimization problem corresponding to the Hodge Laplacian, and we show that this minimization problem is well-posed.
Khademi, Ali & Vatne, Jon Eivind (2020)
Estimation of the interpolation error for semiregular prismatic elements
In this paper we introduce the semiregularity property for a family of decompositions of a polyhedron into a natural class of prisms. In such a family, prismatic elements are allowed to be very flat or very long compared to their triangular bases, and the edges of quadrilateral faces can be nonparallel. Moreover, the triangular faces of each element are either parallel or skew to each other. To estimate the error of the interpolation operator defined on the finite space whose basis functions are defined on the general prismatic elements, we consider quadratic polynomials as the basis functions for that space which are bilinear on the reference prism. We then prove that under this modification of the semiregularity criterion, the interpolation error is of order O(h) in the H1-norm.
The purpose of this article is to develop tools for producing multiple structures on smooth varieties, based on the theory for curves by Banica and Forster [1, 2]. By recursively extending schemes, we show how all Cohen-Macaulay scheme structures of this kind can be found. Similar results have been obtained by Manolache [12, 15-17], using a different recursive construction. The construction in this article complements Manolache's methods, and for some of the applications we have in mind, our construction gives more flexibility [18, 19]. As an application of the theory, we reformulate Hartshorne's Conjecture on complete intersections in codimension two in terms of multiple schemes of degrees two and three.
Vatne, Jon Eivind (2008)
Double structures on rational space curves
281(3) , s. 434- 441.
There are very many non-reduced schemes. In this paper, we consider two examples to back this statement: we give lists of double scheme structures on a twisted cubic, and we construct rank two bundles on projective 3-space with prescribed Chern classes, from double structures on smooth rational curves. (C) 2008 WILEY-VCH Verlag GmbH & Co. KG A, Weinheim.
Fløystad, Gunnar & Vatne, Jon Eivind (2006)
PBW-deformations of N-Koszul algebras
302(1) , s. 116- 155.
Fløystad, Gunnar & Vatne, Jon Eivind (2005)
(Bi)-Cohen-Macaulay simplicial complexes and their associated coherent sheaves
33(9) , s. 3121- 3136.
Vatne, Jon Eivind (2005)
(Bi)-Cohen-Macaulay Simplicial Complexes and Their Associated Coherent Sheaves
33(9) , s. 3121- 3.
Vatne, Jon Eivind & Fløystad, Gunnar (2002)
(Bi-) Cohen-Macaulay simplicial complexes and their associated coherent sheaves
Gulbrandsen, Martin G; Kleppe, Johannes, Kro, Tore August & Vatne, Jon Eivind (2015)
Matematikk for ingeniørfag - oppgaver og fasit
[Textbook].
Gulbrandsen, Martin G; Kleppe, Johannes, Kro, Tore August & Vatne, Jon Eivind (2013)
Matematikk for ingeniørfag
[Textbook].
Mørken, Knut Martin; Simonsen, Ingve, Malthe-Sørensen, Anders, Hammer, Hugo Lewi, Løyning, Terje Brinck, Vatne, Jon Eivind, Nøst, Elisabeth, Dahl, Lars Oswald & Sasaki, nina (2011)
Beregninsorientert utdanning: En veileder for universiteter og høgskoler i Norge
[Report Research].
Mørken, Knut Martin; Malthe-Sørensen, Anders, Simonsen, Ingve, Hammer, Hugo Lewi, Vatne, Jon Eivind, Nøst, Elisabeth, Løyning, Terje Brinck, Dahl, Lars Oswald & Sasaki, Nina (2011)
Computing in Science Education. A guide for universities and colleges in Norway
[Report Research].
Vatne, Jon Eivind (2005)
PBW-deformations of N-Koszul algebras and Their A_\infty Ext Algebras
[Conference Lecture]. Event
Fløystad, Gunnar & Vatne, Jon Eivind (2004)
PBW-deformations of N-Koszul algebras
[Report Research].
Fløystad, Gunnar & Vatne, Jon Eivind (2003)
(Bi)-Cohen-Macaulay simplicial complexes and their associated coherent sheaves
[Conference Lecture]. Event
Vatne, Jon Eivind & Fløystad, Gunnar (2002)
(Bi-) Cohen-Macaulay complexes and their associated coherent sheaves
