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  • 1.
    Araujo-Cabarcas, Juan Carlos
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Engström, Christian
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On spurious solutions in finite element approximations of resonances in open systems2017In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 74, no 10, p. 2385-2402Article in journal (Refereed)
    Abstract [en]

    In this paper, we discuss problems arising when computing resonances with a finite element method. In the pre-asymptotic regime, we detect for the one dimensional case, spurious solutions in finite element computations of resonances when the computational domain is truncated with a perfectly matched layer (PML) as well as with a Dirichlet-to-Neumann map (DtN). The new test is based on the Lippmann–Schwinger equation and we use computations of the pseudospectrum to show that this is a suitable choice. Numerical simulations indicate that the presented test can distinguish between spurious eigenvalues and true eigenvalues also in difficult cases.

  • 2. de Prenter, Frits
    et al.
    Lehrenfeld, Christoph
    Massing, André
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A note on the stability parameter in Nitsche's method for unfitted boundary value problems2018In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 75, no 12, p. 4322-4336Article in journal (Refereed)
    Abstract [en]

    Nitsche's method is a popular approach to implement Dirichlet-type boundary conditions in situations where a strong imposition is either inconvenient or simply not feasible. The method is widely applied in the context of unfitted finite element methods. Of the classical (symmetric) Nitsche's method it is well-known that the stabilization parameter in the method has to be chosen sufficiently large to obtain unique solvability of discrete systems. In this short note we discuss an often used strategy to set the stabilization parameter and describe a possible problem that can arise from this. We show that in specific situations error bounds can deteriorate and give examples of computations where Nitsche's method yields large and even diverging discretization errors. 

  • 3.
    Eklund, Patrik
    et al.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Galán, M.A.
    Umeå University, Faculty of Science and Technology, Department of Computing Science.
    Medina, J.
    Ojeda-Aciego, M.
    Valverde, A.
    Setfunctors, L-fuzzysetcategories, and generalizedterms2002In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 43, no 6-7, p. 693-705Article in journal (Refereed)
    Abstract [en]

    In this work, we generalize previous constructions of fuzzysetcategories, introduced in [1], by consideringL-fuzzysets in which the values of the characteristic functions run on a completely distributive lattice, rather than in the unit real interval. Later, these L-fuzzysets are used to define the L-fuzzycategories, which are proven to be rational. In the final part of the paper, the L-fuzzyfunctors given by the extension principles are provided with a structure of monad which is used, together with the functorial definition of theterm monad, to provide monad compositions as a basis for a notion of generalisedterms.

  • 4.
    Engström, Christian
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Giani, Stefano
    Grubisic, Luka
    Efficient and reliable hp-FEM estimates for quadratic eigenvalue problems and photonic crystal applications2016In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 72, no 4, p. 952-973Article in journal (Refereed)
    Abstract [en]

    We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadratic Fredholm-valued operator functions. Residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem. For the reasons of robustness we also consider approximations of the associated invariant pairs. We show that our estimator inherits the efficiency and reliability properties of the underlying boundary value estimator. As a model problem we consider spectral problems arising in analysis of photonic crystals. In particular, we present an example where a targeted family of eigenvalues cannot be guaranteed to be semisimple. Numerical experiments with hp-FEM show the predicted convergence rates. The measured effectivities of the estimator compare favorably with the performance of the same estimator on the associated boundary value problem. We also present a benchmark estimator, based on the dual weighted residual (DWR) approach, which is more expensive to compute but whose measured effectivities are close to one. 

  • 5.
    Panchuk, Anastasiia
    et al.
    Institute of Mathematics, National Academy of Sciences, Ukraine.
    Puu, Tönu
    Umeå University, Faculty of Social Sciences, Centre for Regional Science (CERUM).
    Stability in a non-autonomous iterative system: an application to oligopoly2009In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 58, no 10, p. 2022-2034Article in journal (Refereed)
    Abstract [en]

    This paper reconsiders the relation between oligopoly and perfect competition, more specifically the problem of emergent instability when the number of competitors increases, as pointed out by several authors. A process of mixed short and long run dynamics is set up. In the short run the competitors are subject to capacity limits due to fixed capital stocks, in the long run they may renew these stocks and so in the moments of reinvestment have access to a constant returns technology. The evolution of the system depends on the number of competitors, the interval between their entry on the market, and the durability of capital. The main result is a theorem showing that if capital has a durability of more periods than the spacing of reinvestment times among the firms, multiplied with their total number, then the system always contracts to the Cournot equilibrium state.

  • 6. Tian, C.
    et al.
    Zhang, Lai
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Hopf bifurcation analysis in a diffusive food-chain model with time delay2013In: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 66, no 10, p. 2139-2153Article in journal (Refereed)
    Abstract [en]

    In this paper, we consider the direction and stability of time-delay induced Hopf bifurcation in a three species food-chain model with diffusion. By means of analyzing eigenvalue spectrum and Lyapunov functional, we show that the positive equilibrium is asymptotically stable in the absence of time delay, but loses its stability via the Hopf bifurcation when the time delay increases beyond a threshold. Using the norm form and the center manifold theory, we investigate the stability and direction of the Hopf bifurcation. The instability of the Hopf bifurcation leads to the emergence of spatial patterns. Numerical calculations are performed to illustrate our theoretical results.

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