| 1. |
- Målqvist, Axel, 1978, et al.
(författare)
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Localization of elliptic multiscale problems
- 2014
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Ingår i: Mathematics of Computation. - 1088-6842 .- 0025-5718. ; 83:290, s. 2583-2603
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Tidskriftsartikel (refereegranskat)abstract
- This paper constructs a local generalized finite element basis for elliptic problems with heterogeneous and highly varying coefficients. The basis functions are solutions of local problems on vertex patches. The error of the corresponding generalized finite element method decays exponentially with respect to the number of layers of elements in the patches. Hence, on a uniform mesh of size $ H$, patches of diameter $ H\log (1/H)$ are sufficient to preserve a linear rate of convergence in $ H$ without pre-asymptotic or resonance effects. The analysis does not rely on regularity of the solution or scale separation in the coefficient. This result motivates new and justifies old classes of variational multiscale methods. - See more at: http://www.ams.org/journals/mcom/2014-83-290/S0025-5718-2014-02868-8/#sthash.z2CCFXIg.dpuf
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| 2. |
- Hakberg, Bengt, 1939
(författare)
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A Discrete KPP-Theory for Fisher's Equation
- 2013
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Ingår i: Mathematics of Computation. - 0025-5718 .- 1088-6842. ; 82:282, s. 781-802
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Tidskriftsartikel (refereegranskat)abstract
- The purpose of this paper is to extend the theory by Kolmogorov, Petrowsky and Piscunov (KPP) for Fisher's equation, to a discrete solution. We approximate the time derivative in Fisher's equation by an explicit Euler scheme and the diffusion operator by a symmetric difference scheme of second order. We prove that the discrete solution converges towards a traveling wave, under restrictions in the time-and space-widths, as the number of time steps increases to infinity. We also prove that the flame velocity can be determined as a solution to an optimization problem.
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| 3. |
- Sheen, D, et al.
(författare)
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A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature
- 2000
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Ingår i: Math. Comp.. - 0025-5718 .- 1088-6842. ; 69:229, s. 177-195
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Tidskriftsartikel (refereegranskat)abstract
- We treat the time discretization of an initial-value problem for a homogeneous abstract parabolic equation by first using a representation of the solution as an integral along the boundary of a sector in the right half of the complex plane, then transforming this into a real integral on the finite interval , and finally applying a standard quadrature formula to this integral. The method requires the solution of a finite set of elliptic problems with complex coefficients, which are independent and may therefore be done in parallel. The method is combined with spatial discretization by finite elements.
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| 4. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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A new approach to Richardson extrapolation in the finite element method for second order elliptic problems
- 2009
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Ingår i: Mathematics of Computation. - 0025-5718 .- 1088-6842. ; 78:4, s. 1951-1973
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Tidskriftsartikel (refereegranskat)abstract
- This paper presents a nonstandard local approach to Richardson extrapolation, when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in $ \mathbb{R}^N$, $ N \ge 2$. The main feature of the approach is that it does not rely on a traditional asymptotic error expansion, but rather depends on a more easily proved weaker a priori estimate, derived in [19], called an asymptotic error expansion inequality. In order to use this inequality to verify that the Richardson procedure works at a point, we require a local condition which links the different subspaces used for extrapolation. Roughly speaking, this condition says that the subspaces are similar about a point, i.e., any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point. Examples of finite element subspaces that occur in practice and satisfy this condition are given.
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| 5. |
- Avelin, Helen
(författare)
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Computations of Eisenstein series on Fuchsian groups
- 2008
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Ingår i: Mathematics of Computation. - 0025-5718 .- 1088-6842. ; 77:263, s. 1779-1800
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Tidskriftsartikel (refereegranskat)abstract
- We present numerical investigations of the value distribution and distribution of Fourier coefficients of the Eisenstein series E(z, s) on arithmetic and non-arithmetic Fuchsian groups. Our numerics indicate a Gaussian limit value distribution for a real- valued rotation of E(z; s) as Res = 1/2, Im s -> , infinity and also, on non-arithmetic groups, a complex Gaussian limit distribution for E(z, s) when Res > , 1/2 near 1/2 and Im s -> , infinity at least if we allow Re s. 1/2 at some rate. Furthermore, on non-arithmetic groups and for fixed s with Re s = 1/2 near 1/2, our numerics indicate a Gaussian limit distribution for the appropriately normalized Fourier coefficients.
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| 6. |
- Avelin, Helen
(författare)
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Deformation of Γ0(5)-cusp forms
- 2007
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Ingår i: Mathematics of Computation. - 0025-5718 .- 1088-6842. ; 76:257, s. 361-384
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Tidskriftsartikel (refereegranskat)abstract
- We develop an algorithm for numerical computation of the Eisenstein series on a Riemann surface of constant negative curvature. We focus in particular on the computation of the poles of the Eisenstein series. Using our numerical methods, we study the spectrum of the Laplace-Beltrami operator as the surface is being deformed. Numerical evidence of the destruction of $ \Gamma_0(5)$-cusp forms is presented.
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| 7. |
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| 8. |
- Björn, Anders, et al.
(författare)
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Factors of generalized fermat numbers
- 1998
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Ingår i: Mathematics of Computation. - 0025-5718 .- 1088-6842. ; 67:221, s. 441-446
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Tidskriftsartikel (refereegranskat)
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| 9. |
- Björn, Anders, et al.
(författare)
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FACTORS OF GENERALIZED FERMAT NUMBERS (vol 67, pg 441, 1998) : Table errata 2
- 2011
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Ingår i: Mathematics of Computation. - 0025-5718 .- 1088-6842. ; 80:275, s. 1865-1866
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Tidskriftsartikel (refereegranskat)abstract
- We note that one more factor is missing from Table 1 in Bjorn-Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), 441 446, in addition to the three already reported upon in Bjorn-Riesel, Table errata to "Factors of generalized Fermat numbers", Math. Comp. 74 (2005), p. 2099.
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| 10. |
- Björn, Anders, et al.
(författare)
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Factors of generalized fermat numbers (vol 67, pg 441, 1998)
- 2005
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Ingår i: Mathematics of Computation. - 0025-5718 .- 1088-6842. ; 74:252, s. 2099-2099
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Tidskriftsartikel (refereegranskat)abstract
- We note that three factors are missing from Table 1 in Factors of generalized Fermat numbers by A. Bjorn and H. Riesel published in Math. Comp. 67 (1998), 441-446.
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