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  • 1.
    Abrahamsson, Leif R.
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för teknisk databehandling. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
    Keller, Herbert B.
    Kreiss, Heinz-Otto
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för teknisk databehandling. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
    Difference approximations for singular perturbations of systems of ordinary differential equations1974Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 22, s. 367-391Artikel i tidskrift (Refereegranskat)
  • 2. Babuska, I.
    et al.
    Nobile, F.
    Tempone, Raul
    Worst case scenario analysis for elliptic problems with uncertainty2005Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 101, nr 2, s. 185-219Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This work studies linear elliptic problems under uncertainty. The major emphasis is on the deterministic treatment of such uncertainty. In particular, this work uses the Worst Scenario approach for the characterization of uncertainty on functional outputs (quantities of physical interest). Assuming that the input data belong to a given functional set, eventually infinitely dimensional, this work proposes numerical methods to approximate the corresponding uncertainty intervals for the quantities of interest. Numerical experiments illustrate the performance of the proposed methodology.

  • 3. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, M. G.
    Zahedi, Sara
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
    Cut finite element methods for coupled bulk–surface problems2016Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 133, nr 2, s. 203-231Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We develop a cut finite element method for a second order elliptic coupled bulk-surface model problem. We prove a priori estimates for the energy and (Formula presented.) norms of the error. Using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method. Finally, we present a numerical example illustrating the accuracy and the robustness of our approach.

  • 4.
    Burman, Erik
    et al.
    University College London, UK, Department of Mathematics.
    Hansbo, Peter
    Jönköping University, School of Engineering, JTH, Product Development.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larsson, Karl
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Massing, Andre
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Finite element approximation of the Laplace-Beltrami operator on a surface with boundary2019Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 141, nr 1, s. 141-172Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We develop a finite element method for the Laplace–Beltrami operator on a surface with boundary and nonhomogeneous Dirichlet boundary conditions. The method is based on a triangulation of the surface and the boundary conditions are enforced weakly using Nitsche's method. We prove optimal order a priori error estimates for piecewise continuous polynomials of order k ≥ 1 in the energy and L2 norms that take the approximation of the surface and the boundary into account.

  • 5.
    Burman, Erik
    et al.
    Department of Mathematics, University College London, London, United Kingdom.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Material och tillverkning.
    Larson, Mats G.
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Larsson, Karl
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Massing, André
    Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.
    Finite element approximation of the Laplace–Beltrami operator on a surface with boundary2019Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 141, nr 1, s. 141-172Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We develop a finite element method for the Laplace–Beltrami operator on a surface with boundary and nonhomogeneous Dirichlet boundary conditions. The method is based on a triangulation of the surface and the boundary conditions are enforced weakly using Nitsche’s method. We prove optimal order a priori error estimates for piecewise continuous polynomials of order (Formula presented.) in the energy and (Formula presented.) norms that take the approximation of the surface and the boundary into account. 

  • 6. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Zahedi, Sara
    Cut finite element methods for coupled bulk-surface problems2016Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 133, nr 2, s. 203-231Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We develop a cut finite element method for a second order elliptic coupled bulk-surface model problem. We prove a priori estimates for the energy and norms of the error. Using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method. Finally, we present a numerical example illustrating the accuracy and the robustness of our approach.

  • 7.
    Burman, Erik
    et al.
    University College London.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering. Högskolan i Jönköping, Tekniska Högskolan, JTH, Produktutveckling.
    Larson, Mats G.
    Umeå University.
    Zahedi, Sara
    KTH Royal Institute of Technology.
    Cut finite element methods for coupled bulk–surface problems2016Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 133, nr 2, s. 203-231Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We develop a cut finite element method for a second order elliptic coupled bulk-surface model problem. We prove a priori estimates for the energy and L2 norms of the error. Using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method. Finally, we present a numerical example illustrating the accuracy and the robustness of our approach.

  • 8. Burman, Erik
    et al.
    Hansbo, Peter
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Zahedi, Sara
    Stabilized CutFEM for the convection problem on surfaces2019Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 141, nr 1, s. 103-139Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We develop a stabilized cut finite element method for the convection problem on a surface based on continuous piecewise linear approximation and gradient jump stabilization terms. The discrete piecewise linear surface cuts through a background mesh consisting of tetrahedra in an arbitrary way and the finite element space consists of piecewise linear continuous functions defined on the background mesh. The variational form involves integrals on the surface and the gradient jump stabilization term is defined on the full faces of the tetrahedra. The stabilization term serves two purposes: first the method is stabilized and secondly the resulting linear system of equations is algebraically stable. We establish stability results that are analogous to the standard meshed flat case and prove h3/2 order convergence in the natural norm associated with the method and that the full gradient enjoys h3/4 order of convergence in L2. We also show that the condition number of the stiffness matrix is bounded by h−2. Finally, our results are verified by numerical examples.

  • 9.
    Burman, Erik
    et al.
    UCL, Department of Mathematics, London, United Kingdom.
    Hansbo, Peter
    Högskolan i Jönköping, Tekniska Högskolan, JTH, Material och tillverkning.
    Larson, Mats G.
    Umeå Universitet, Department of Mathematics and Mathematical Statistics, Umeå, Sweden.
    Zahedi, Sara
    The Royal Institute of Technology (KTH), Department of Mathematics, Stockholm, Sweden.
    Stabilized CutFEM for the convection problem on surfaces2019Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 141, nr 1, s. 103-139Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We develop a stabilized cut finite element method for the convection problem on a surface based on continuous piecewise linear approximation and gradient jump stabilization terms. The discrete piecewise linear surface cuts through a background mesh consisting of tetrahedra in an arbitrary way and the finite element space consists of piecewise linear continuous functions defined on the background mesh. The variational form involves integrals on the surface and the gradient jump stabilization term is defined on the full faces of the tetrahedra. The stabilization term serves two purposes: first the method is stabilized and secondly the resulting linear system of equations is algebraically stable. We establish stability results that are analogous to the standard meshed flat case and prove h3 / 2 order convergence in the natural norm associated with the method and that the full gradient enjoys h3 / 4 order of convergence in L2. We also show that the condition number of the stiffness matrix is bounded by h- 2. Finally, our results are verified by numerical examples. 

  • 10.
    Burman, Erik
    et al.
    UCL, Dept Math, London WC1E 6BT, England..
    Hansbo, Peter
    Jonkoping Univ, Dept Mech Engn, S-55111 Jonkoping, Sweden..
    Larson, Mats G.
    Umea Univ, Dept Math & Math Stat, S-90187 Umea, Sweden..
    Zahedi, Sara
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Stabilized CutFEM for the convection problem on surfaces2019Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 141, nr 1, s. 103-139Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We develop a stabilized cut finite element method for the convection problem on a surface based on continuous piecewise linear approximation and gradient jump stabilization terms. The discrete piecewise linear surface cuts through a background mesh consisting of tetrahedra in an arbitrary way and the finite element space consists of piecewise linear continuous functions defined on the background mesh. The variational form involves integrals on the surface and the gradient jump stabilization term is defined on the full faces of the tetrahedra. The stabilization term serves two purposes: first the method is stabilized and secondly the resulting linear system of equations is algebraically stable. We establish stability results that are analogous to the standard meshed flat case and prove h3/2 order convergence in the natural norm associated with the method and that the full gradient enjoys h3/4 order of convergence in L2. We also show that the condition number of the stiffness matrix is bounded by h-2. Finally, our results are verified by numerical examples.

  • 11.
    Cohen, David
    et al.
    Department of Mathematical Sciences, NTNU.
    Hairer, Ernst
    Section de Mathématiques, Université de Genève.
    Lubich, Christian
    Mathematisches Institut, Universität Tübingen.
    Conservation of energy, momentum and actions in numerical discretizations of non-linear wave equations2008Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 110, nr 2, s. 113-143Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    For classes of symplectic and symmetric time-stepping methods- trigonometric integrators and the Stormer-Verlet or leapfrog method-applied to spectral semi-discretizations of semilinear wave equations in a weakly non-linear setting, it is shown that energy, momentum, and all harmonic actions are approximately preserved over long times. For the case of interest where the CFL number is not a small parameter, such results are outside the reach of standard backward error analysis. Here, they are instead obtained via a modulated Fourier expansion in time.

  • 12.
    Cohen, David
    et al.
    Mathematisches Institut, Universität Basel.
    Raynaud, Xavier
    CMA, University of Oslo.
    Convergent numerical schemes for the compressible hyperelastic rod wave equation2012Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 122, s. 1-59Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We propose a fully discretised numerical scheme for the hyperelastic rod wave equation on the line. The convergence of the method is established. Moreover, the scheme can handle the blow-up of the derivative which naturally occurs for this equation. By using a time splitting integrator which preserves the invariants of the problem, we can also show that the scheme preserves the positivity of the energy density.

  • 13.
    Cohen, David
    et al.
    Department of mathematics, University of Basel.
    Sigg, Magdalena
    Department of mathematics, University of Basel.
    Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations2012Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 121, nr 1, s. 1-29Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study a class of numerical methods for a system of second-order SDE driven by a linear fast force generating high frequency oscillatory solutions. The proposed schemes permit the use of large step sizes, have uniform global error bounds in the position (i.e. independent of the large frequencies present in the SDE) and offer various additional properties. This new family of numerical integrators for SDE can be viewed as a stochastic generalisation of the trigonometric integrators for highly oscillatory deterministic problems.

  • 14.
    Elfverson, Daniel
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Avdelningen för beräkningsvetenskap. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
    Ginting, Victor
    Henning, Patrick
    On multiscale methods in Petrov–Galerkin formulation2015Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 131, s. 643-682Artikel i tidskrift (Refereegranskat)
  • 15.
    Engström, Christian
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Spectral approximation of quadratic operator polynomials arising in photonic band structure calculations2014Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 126, nr 3, s. 413-440Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Galerkin spectral approximation theory for non-self-adjoint quadratic operator polynomials with periodic coefficients is considered. The main applications are complex band structure calculations in metallic photonic crystals, periodic waveguides, and metamaterials. We show that the spectrum of the considered operator polynomials consists of isolated eigenvalues of finite multiplicity with a nonzero imaginary part. The spectral problem is equivalent to a non-compact block operator matrix and norm convergence is shown for a block operator matrix having the same generalized eigenvectors as the original operator. Convergence rates of finite element discretizations are considered and numerical experiments with the p -version and the h -version of the finite element method confirm the theoretical convergence rates.

  • 16.
    Engström, Christian
    Umeå university, Sweden.
    Spectral approximation of quadratic operator polynomials arising in photonic band structure calculations2014Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 126, nr 3, s. 413-440Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Galerkin spectral approximation theory for non-self-adjoint quadratic operator polynomials with periodic coefficients is considered. The main applications are complex band structure calculations in metallic photonic crystals, periodic waveguides, and metamaterials. We show that the spectrum of the considered operator polynomials consists of isolated eigenvalues of finite multiplicity with a nonzero imaginary part. The spectral problem is equivalent to a non-compact block operator matrix and norm convergence is shown for a block operator matrix having the same generalized eigenvectors as the original operator. Convergence rates of finite element discretizations are considered and numerical experiments with the p -version and the h -version of the finite element method confirm the theoretical convergence rates.

  • 17.
    Gustafsson, Bertil
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för teknisk databehandling. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
    A numerical method for solving singular boundary value problems1973Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 21, s. 328-344Artikel i tidskrift (Refereegranskat)
  • 18.
    Hansbo, Peter
    et al.
    Högskolan i Jönköping, Tekniska Högskolan, JTH. Forskningsmiljö Produktutveckling - Simulering och optimering.
    Lovadina, Carlo
    Perugia, Ilaria
    Sangalli, Giancarlo
    A Lagrange multiplier method for the finite element solution of elliptic interface problems using non-matching meshes2005Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 100, nr 1, s. 91-115Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we propose a Lagrange multiplier method for the finite element solution of multi-domain elliptic partial differential equations using non-matching meshes. The interface Lagrange multiplier is chosen with the purpose of avoiding the cumbersome integration of products of functions on unrelated meshes (e.g, we will consider global polynomials as multiplier). The ideas are illustrated using Poissons equation as a model, and the proposed method is shown to be stable and optimally convergent. Numerical experiments demonstrating the theoretical results are also presented.

  • 19.
    Hemmingsson, Lina
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för teknisk databehandling. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
    A semi-circulant preconditioner for the convection-diffusion equation1998Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 81, s. 211-248Artikel i tidskrift (Refereegranskat)
  • 20.
    Henning, Patrick
    et al.
    University of Münster, Germany.
    Ohlberger, Mario
    University of Münster, Germany.
    The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains2009Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 113, nr 4, s. 601-629Artikel i tidskrift (Refereegranskat)
  • 21.
    Holm, Bärbel
    et al.
    KTH, Skolan för elektroteknik och datavetenskap (EECS), Beräkningsvetenskap och beräkningsteknik (CST).
    Wihler, Thomas P.
    Univ Bern, Math Inst, Sidlerstr 5, CH-3012 Bern, Switzerland..
    Continuous and discontinuous Galerkin time stepping methods for nonlinear initial value problems with application to finite time blow-up2018Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 138, nr 3, s. 767-799Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider continuous and discontinuous Galerkin time stepping methods of arbitrary order as applied to first-order initial value ordinary differential equation problems in real Hilbert spaces. Our only assumption is that the nonlinearities are continuous; in particular, we include the case of unbounded nonlinear operators. Specifically, we develop new techniques to prove general Peano-type existence results for discrete solutions. In particular, our results show that the existence of solutions is independent of the local approximation order, and only requires the local time steps to be sufficiently small (independent of the polynomial degree). The uniqueness of (local) solutions is addressed as well. In addition, our theory is applied to finite time blow-up problems with nonlinearities of algebraic growth. For such problems we develop a time step selection algorithm for the purpose of numerically computing the blow-up time, and provide a convergence result.

  • 22.
    Häggblad, Jon
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Runborg, Olof
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30). KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Accuracy of staircase approximations in finite-difference methods for wave propagation2014Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 128, nr 4, s. 741-771Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    While a number of increasingly sophisticated numerical methods have been developed for time-dependent problems in electromagnetics, the Yee scheme is still widely used in the applied fields, mainly due to its simplicity and computational efficiency. A fundamental drawback of the method is the use of staircase boundary approximations, giving inconsistent results. Usually experience of numerical experiments provides guidance of the impact of these errors on the final simulation result. In this paper, we derive exact discrete solutions to the Yee scheme close to the staircase approximated boundary, enabling a detailed theoretical study of the amplitude, phase and frequency errors created. Furthermore, we show how evanescent waves of amplitude occur along the boundary. These characterize the inconsistencies observed in electromagnetic simulations and the locality of the waves explain why, in practice, the Yee scheme works as well as it does. The analysis is supported by detailed proofs and numerical examples.

  • 23.
    Jarlebring, Elias
    et al.
    KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30).
    Michiels, Wim
    Meerbergen, Karl
    A linear eigenvalue algorithm for the nonlinear eigenvalue problem2012Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 122, nr 1, s. 169-195Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. The first result of this paper is a characterization of the solutions to an arbitrary (analytic) nonlinear eigenvalue problem (NEP) as the reciprocal eigenvalues of an infinite dimensional operator denoted . We consider the Arnoldi method for the operator and show that with a particular choice of starting function and a particular choice of scalar product, the structure of the operator can be exploited in a very effective way. The structure of the operator is such that when the Arnoldi method is started with a constant function, the iterates will be polynomials. For a large class of NEPs, we show that we can carry out the infinite dimensional Arnoldi algorithm for the operator in arithmetic based on standard linear algebra operations on vectors and matrices of finite size. This is achieved by representing the polynomials by vector coefficients. The resulting algorithm is by construction such that it is completely equivalent to the standard Arnoldi method and also inherits many of its attractive properties, which are illustrated with examples.

  • 24. Johansson, August
    et al.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary2013Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 123, nr 4, s. 607-628Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We present a discontinuous Galerkin method, based on the classical method of Nitsche, for elliptic problems with an immersed boundary representation on a structured grid. In such methods very small elements typically occur at the boundary, leading to breakdown of the discrete coercivity as well as numerical instabilities. In this work we propose a method that avoids using very small elements on the boundary by associating them to a neighboring element with a sufficiently large intersection with the domain. This construction allows us to prove the crucial inverse inequality that leads to a coercive bilinear form and as a consequence we obtain optimal order a priori error estimates. Furthermore, we prove a bound of the condition number of the stiffness matrix. All the results are valid for polynomials of arbitrary order. We also discuss the implementation of the method and present numerical examples in three dimensions.

  • 25.
    Kreiss, Heinz-Otto
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för teknisk databehandling. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
    On the numerical solution of the spherically symmetric diffusion equation1968Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 12, s. 223-225Artikel i tidskrift (Refereegranskat)
  • 26.
    Kumar, Kundan
    et al.
    University of Texas at Austin, Austin, USA.
    Pop, I. S.
    Eindhoven University of TechnologyEindhovenThe Netherlands; University of Bergen, Bergen, Norway.
    Radu, F. A.
    University of Bergen, Bergen, Norway.
    Convergence analysis for a conformal discretization of a model for precipitation and dissolution in porous media2014Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 127, nr 4, s. 715-749Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we discuss the numerical analysis of an upscaled (core scale) model describing the transport, precipitation and dissolution of solutes in a porous medium. The particularity lies in the modeling of the reaction term, especially the dissolution term, which has a multivalued character. We consider the weak formulation for the upscaled equation and provide rigorous stability and convergence results for both the semi-discrete (time discretization) and the fully discrete schemes. In doing so, compactness arguments are employed.

  • 27.
    Larson, Mats G.
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Malqvist, Axel
    A posteriori error estimates for mixed finite element approximations of parabolic problems2011Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 118, nr 1, s. 33-48Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We derive residual based a posteriori error estimates for parabolic problems on mixed form solved using Raviart-Thomas-Nedelec finite elements in space and backward Euler in time. The error norm considered is the flux part of the energy, i.e. weighted L (2)(Omega) norm integrated over time. In order to get an optimal order bound, an elementwise computable post-processed approximation of the scalar variable needs to be used. This is a common technique used for elliptic problems. The final bound consists of terms, capturing the spatial discretization error and the time discretization error and can be used to drive an adaptive algorithm.

  • 28. Larson, Mats G.
    et al.
    Målqvist, Axel
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Avdelningen för teknisk databehandling. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
    A posteriori error estimates for mixed finite element approximations of elliptic problems2008Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 108, s. 487-500Artikel i tidskrift (Refereegranskat)
  • 29.
    Larson, Mats G.
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Målqvist, Axel
    A posteriori error estimates for mixed finite element approximations of elliptic problems2008Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 108, nr 3, s. 487-500Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We derive residual based a posteriori error estimates of the flux in L 2-norm for a general class of mixed methods for elliptic problems. The estimate is applicable to standard mixed methods such as the Raviart–Thomas–Nedelec and Brezzi–Douglas–Marini elements, as well as stabilized methods such as the Galerkin-Least squares method. The element residual in the estimate employs an elementwise computable postprocessed approximation of the displacement which gives optimal order.

  • 30. Larson, Mats G.
    et al.
    Målqvist, Axel
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Avdelningen för teknisk databehandling. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
    A posteriori error estimates for mixed finite element approximations of parabolic problems2011Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 118, s. 33-48Artikel i tidskrift (Refereegranskat)
  • 31.
    Larsson, Karl
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Larson, Mats G
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Continuous piecewise linear finite elements for the Kirchhoff–Love plate equation2012Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 121, nr 1, s. 65-97Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A family of continuous piecewise linear finite elements for thin plate problems is presented. We use standard linear interpolation of the deflection field to reconstruct a discontinuous piecewise quadratic deflection field. This allows us to use discontinuous Galerkin methods for the Kirchhoff–Love plate equation. Three example reconstructions of quadratic functions from linear interpolation triangles are presented: a reconstruction using Morley basis functions, a fully quadratic reconstruction, and a more general least squares approach to a fully quadratic reconstruction. The Morley reconstruction is shown to be equivalent to the basic plate triangle (BPT). Given a condition on the reconstruction operator, a priori error estimates are proved in energy norm and L2 norm. Numerical results indicate that the Morley reconstruction/BPT does not converge on unstructured meshes while the fully quadratic reconstruction show optimal convergence.

  • 32.
    Massing, Andre
    et al.
    Simula Research Laboratory, Oslo, Norway.
    Larson, Mats G.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
    Logg, Anders
    Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden.
    Rognes, Marie E.
    Simula Research Laboratory, Oslo, Norway.
    A stabilized Nitsche overlapping mesh method for the Stokes problem2014Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 128, nr 1, s. 73-101Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We develop a Nitsche-based formulation for a general class of stabilized finite element methods for the Stokes problem posed on a pair of overlapping, non-matching meshes. By extending the least-squares stabilization to the overlap region, we prove that the method is stable, consistent, and optimally convergent. To avoid an ill-conditioned linear algebra system, the scheme is augmented by a least-squares term measuring the discontinuity of the solution in the overlap region of the two meshes. As a consequence, we may prove an estimate for the condition number of the resulting stiffness matrix that is independent of the location of the interface. Finally, we present numerical examples in three spatial dimensions illustrating and confirming the theoretical results.

  • 33. Moon, K. S.
    et al.
    Szepessy, Anders
    KTH, Tidigare Institutioner                               , Numerisk analys och datalogi, NADA.
    Tempone, Raul
    KTH, Tidigare Institutioner                               , Numerisk analys och datalogi, NADA.
    Zouraris, G. E.
    Convergence rates for adaptive approximation of ordinary differential equations2003Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 96, nr 1, s. 99-129Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This paper constructs an adaptive algorithm for ordinary differential equations and analyzes its asymptotic behavior as the error tolerance parameter tends to zero. An adaptive algorithm, based on the error indicators and successive subdivision of time steps, is proven to stop with the optimal number, N, of steps up to a problem independent factor defined in the algorithm. A version of the algorithm with decreasing tolerance also stops with the total number of steps, including all refinement levels, bounded by O(N). The alternative version with constant tolerance stops with O(N log N) total steps. The global error is bounded by the tolerance parameter asymptotically as the tolerance tends to zero. For a p-th order accurate method the optimal number of adaptive steps is proportional to the p-th root of the L 1/p+1 quasi-norm of the error density, while the number of uniform steps, with the same error, is proportional to the p-th root of the larger L-1-norm of the error density.

  • 34. Moon, K. S.
    et al.
    Szepessy, Anders
    KTH, Tidigare Institutioner, Numerisk analys och datalogi, NADA.
    Tempone, Raúl
    Zouraris, G. E.
    A variational principle for adaptive approximation of ordinary differential equations2003Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 96, nr 1, s. 131-152Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A variational principle, inspired by optimal control, yields a simple derivation of an error representation, global error = Sigma local error . weight, for general approximation of functions of solutions to ordinary differential equations. This error representation is then approximated by a sum of computable error indicators, to obtain a useful global error indicator for adaptive mesh refinements. A uniqueness formulation is provided for desirable error representations of adaptive algorithms.

  • 35.
    Runborg, Olof
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA. KTH, Centra, SeRC - Swedish e-Science Research Centre.
    Analysis of high order fast interface tracking methods2014Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 128, nr 2, s. 339-375Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Fast high order methods for the propagation of an interface in a velocity field are constructed and analyzed. The methods are generalizations of the fast interface tracking method proposed in Runborg (Commun Math Sci 7:365-398, 2009). They are based on high order subdivision to make a multiresolution decomposition of the interface. Instead of tracking marker points on the interface the related wavelet vectors are tracked. Like the markers they satisfy ordinary differential equations (ODEs), but fine scale wavelets can be tracked with longer timesteps than coarse scale wavelets. This leads to methods with a computational cost of rather than for markers and reference timestep . These methods are proved to still have the same order of accuracy as the underlying direct ODE solver under a stability condition in terms of the order of the subdivision, the order of the ODE solver and the time step ratio between wavelet levels. In particular it is shown that with a suitable high order subdivision scheme any explicit Runge-Kutta method can be used. Numerical examples supporting the theory are also presented.

  • 36.
    Starius, Göran
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för teknisk databehandling. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
    Composite mesh difference methods for elliptic boundary value problems1977Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 28, s. 243-258Artikel i tidskrift (Refereegranskat)
  • 37.
    Starius, Göran
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för teknisk databehandling. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
    Constructing orthogonal curvilinear meshes by solving initial value problems1977Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 28, s. 25-48Artikel i tidskrift (Refereegranskat)
  • 38.
    Starius, Göran
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för teknisk databehandling. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Institutionen för informationsteknologi, Numerisk analys.
    On composite mesh difference methods for hyperbolic differential equations1980Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 35, s. 241-255Artikel i tidskrift (Refereegranskat)
  • 39.
    Sydow, Bengt von
    Luleå tekniska universitet.
    Error estimates for Gaussian quadrature formulae1977Ingår i: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 29, nr 1, s. 59-64Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The author derives both strict and asymptotic error bounds for the Gauss-Jacobi quadrature formula with respect to a general measure. The estimates involve the maximum modulus of the integrand on a contour in the complex plane. The methods are elementary complex analysis

1 - 39 av 39
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