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  • 1.
    Anton, Rikard
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Cohen, David
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Univ Innsbruck, Dept Math, Innsbruck, Austria.
    Larsson, Stig
    Wang, Xiaojie
    Full discretization of semilinear stochastic wave equations driven by multiplicative noise2016Inngår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 54, nr 2, s. 1093-1119Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A fully discrete approximation of the semilinear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space, and a stochastic trigonometric method is used for the temporal approximation. This explicit time integrator allows for mean-square error bounds independent of the space discretization and thus does not suffer from a step size restriction as in the often used Stormer-Verlet leapfrog scheme. Furthermore, it satisfies an almost trace formula (i.e., a linear drift of the expected value of the energy of the problem). Numerical experiments are presented and confirm the theoretical results.

  • 2.
    Berggren, Martin
    Department of Information Technology, Uppsala University.
    Approximations of very weak solutions to boundary-value problems2004Inngår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 42, nr 2, s. 860-877Artikkel i tidsskrift (Fagfellevurdert)
  • 3.
    Berggren, Martin
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Kasolis, Fotios
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Weak material approximation of holes with traction-free boundaries2012Inngår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 50, nr 4, s. 1827-1848Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Consider the solution of a boundary-value problem for steady linear elasticity in which the computational domain contains one or several holes with traction-free boundaries. The presence of holes in the material can be approximated using a weak material; that is, the relative density of material rho is set to 0 < epsilon = rho << 1 in the hole region. The weak material approach is a standard technique in the so-called material distribution approach to topology optimization, in which the inhomogeneous relative density of material is designated as the design variable in order to optimize the spatial distribution of material. The use of a weak material ensures that the elasticity problem is uniquely solvable for each admissible value rho is an element of [epsilon, 1] of the design variable. A finite-element approximation of the boundary-value problem in which the weak material approximation is used in the hole regions can be viewed as a nonconforming but convergent approximation of a version of the original problem in which the solution is continuously and elastically extended into the holes. The error in this approximation can be bounded by two terms that depend on epsilon. One term scales linearly with epsilon with a constant that is independent of the mesh size parameter h but that depends on the surface traction required to fit elastic material in the deformed holes. The other term scales like epsilon(1/2) times the finite-element approximation error inside the hole. The condition number of the weak material stiffness matrix scales like epsilon(-1), but the use of a suitable left preconditioner yields a matrix with a condition number that is bounded independently of epsilon. Moreover, the preconditioned matrix admits the limit value epsilon -> 0, and the solution of corresponding system of equations yields in the limit a finite-element approximation of the continuously and elastically extended problem.

    Fulltekst (pdf)
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  • 4.
    Brännström, Åke
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Carlsson, Linus
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Int Inst Appl Syst Anal, Evolut & Ecol Program, A-2361 Laxenburg, Austria.
    Simpson, Daniel
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Int Inst Appl Syst Anal, Evolut & Ecol Program, A-2361 Laxenburg, Austria.
    On the convergence of the Escalator Boxcar Train2013Inngår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 51, nr 6, s. 3213-3231Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The Escalator Boxcar Train (EBT) is a numerical method that is widely used in theoretical biology to investigate the dynamics of physiologically structured population models, i.e., models in which individuals differ by size or other physiological characteristics. The method was developed more than two decades ago, but has so far resisted attempts to give a formal proof of convergence. Using a modern framework of measure-valued solutions, we investigate the EBT method and show that the sequence of approximating solution measures generated by the EBT method converges weakly to the true solution measure under weak conditions on the growth rate, birth rate, and mortality rate. In rigorously establishing the convergence of the EBT method, our results pave the way for wider acceptance of the EBT method beyond theoretical biology and constitutes an important step towards integration with established numerical schemes.

    Fulltekst (pdf)
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  • 5. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    The Penalty-Free Nitsche Method and Nonconforming Finite Elements for the Signorini Problem2017Inngår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 55, nr 6, s. 2523-2539Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We design and analyse a Nitsche method for contact problems. Compared to the seminal work of Chouly and Hild [SIAM J. Numer. Anal., 51 ( 2013), pp. 1295-1307], our method is constructed by expressing the contact conditions in a nonlinear function for the displacement variable instead of the lateral forces. The contact condition is then imposed using the nonsymmetric variant of Nitsche's method that does not require a penalty term for stability. Nonconforming piecewise affine elements are considered for the bulk discretization. We prove optimal error estimates in the energy norm.

  • 6. Burman, Erik
    et al.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Oksanen, Lauri
    Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem2018Inngår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 56, nr 6, s. 3480-3509Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    consider primal-dual mixed finite element methods for the solution of the elliptic Cauchy problem, or other related data assimilation problems. The method has a local conservation property. We derive a priori error estimates using known conditional stability estimates and determine the minimal amount of weakly consistent stabilization and Tikhonov regularization that yields optimal convergence for smooth exact solutions. The effect of perturbations in data is also accounted for. A reduced version of the method, obtained by choosing a special stabilization of the dual variable, can be viewed as a variant of the least squares mixed finite element method introduced by Darde, Hannukainen, and Hyvonen in [SIAM T. Numer. Anal., 51 (2013), pp. 2123-2148]. The main difference is that our choice of regularization does not depend on auxiliary parameters, the mesh size being the only asymptotic parameter. Finally, we show that the reduced method can be used for defect correction iteration to determine the solution of the full method. The theory is illustrated by some numerical examples.

  • 7.
    Cohen, David
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larsson, Stig
    Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg.
    Sigg, Magdalena
    Mathematisches Institut, Universität Basel.
    A trigonometric method for the linear stochastic wave equation2013Inngår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 51, nr 1, s. 204-222Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A fully discrete approximation of the linear stochastic wave equation driven by additive noise is presented. A standard finite element method is used for the spatial discretization and a stochastic trigonometric scheme for the temporal approximation. This explicit time integrator allows for error bounds independent of the space discretization and thus does not have a step-size restriction as in the often used Störmer--Verlet-leap-frog scheme. Moreover, it enjoys a trace formula as does the exact solution of our problem. These favorable properties are demonstrated with numerical experiments. Read More: http://epubs.siam.org/doi/abs/10.1137/12087030X

  • 8.
    Fotios, Kasolis
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Wadbro, Eddie
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Berggren, Martin
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Analysis of fictitious domain approximations of hard scatterers2015Inngår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 53, nr 5, s. 2347-2362Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Consider the Helmholtz equation del center dot alpha del p+k(2 alpha)p = 0 in a domain that contains a so-called hard scatterer. The scatterer is represented by the value alpha = epsilon, for 0 < epsilon << 1, whereas alpha = 1 whenever the scatterer is absent. This scatterer model is often used for the purpose of design optimization and constitutes a fictitious domain approximation of a body characterized by homogeneous Neumann conditions on its boundary. However, such an approximation results in spurious resonances inside the scatterer at certain frequencies and causes, after discretization, ill-conditioned system matrices. Here, we present a stabilization strategy that removes these resonances. Furthermore, we prove that, in the limit epsilon -> 0, the stabilized problem provides linearly convergent approximations of the solution to the problem with an exactly modeled scatterer. Numerical experiments indicate that a finite element approximation of the stabilized problem is free from internal resonances, and they also suggest that the convergence rate is indeed linear with respect to epsilon.

  • 9. Hansbo, Peter
    et al.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Zahedi, Sara
    STABILIZED FINITE ELEMENT APPROXIMATION OF THE MEAN CURVATURE VECTOR ON CLOSED SURFACES2015Inngår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 53, nr 4, s. 1806-1832Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The mean curvature vector of a surface is obtained by letting the Laplace-Beltrami operator act on the embedding of the surface in R-3. In this contribution we develop a stabilized finite element approximation of the mean curvature vector of certain piecewise linear surfaces which enjoys first order convergence in L-2. The stabilization involves the jump in the tangent gradient in the direction of the outer co-normals at each edge in the surface mesh. We consider both standard meshed surfaces and so-called cut surfaces that are level sets of piecewise linear distance functions. We prove a priori error estimates and verify the theoretical results numerically.

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