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| 2. |
- Almanasreh, Hasan, 1981, et al.
(författare)
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Stabilized finite element method for the radial Dirac equation
- 2013
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Ingår i: Journal of Computational Physics. - : Elsevier BV. - 0021-9991 .- 1090-2716. ; 236, s. 426-442
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Tidskriftsartikel (refereegranskat)abstract
- A challenging difficulty in solving the radial Dirac eigenvalue problem numerically is the presence of spurious (unphysical) eigenvalues, among the genuine ones, that are neither related to mathematical interpretations nor to physical explanations. Many attempts have been made and several numerical methods have been applied to solve the problem using the finite element method (FEM), the finite difference method, or other numerical schemes. Unfortunately most of these attempts failed to overcome the difficulty. As a FEM approach, this work can be regarded as a first promising scheme to solve the spuriosity problem com- pletely. Our approach is based on an appropriate choice of trial and test function spaces. We develop a Streamline Upwind Petrov–Galerkin method to the equation and derive an explicit stability parameter.
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| 3. |
- Coombes, S., et al.
(författare)
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Waves in Random Neural Media
- 2012
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Ingår i: Discrete and Continuous Dynamical Systems. - : American Institute of Mathematical Sciences (AIMS). - 1078-0947 .- 1553-5231. ; 32:8, s. 2951-2970
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Tidskriftsartikel (refereegranskat)abstract
- Translationally invariant integro-differential equations are a common choice of model in neuroscience for describing the coarse-grained dynamics of cortical tissue. Here we analyse the propagation of travelling wavefronts in models of neural media that incorporate some form of modulation or randomness such that translational invariance is broken. We begin with a study of neural architectures in which there is a periodic modulation of the neuronal connections. Recent techniques from two-scale convergence analysis are used to construct a homogenized model in the limit that the spatial modulation is rapid compared with the scale of the long range connections. For the special case that the neuronal firing rate is a Heaviside we calculate the speed of a travelling homogenized front exactly and find how the wave speed changes as a function of the amplitude of the modulation. For this special case we further show how to obtain more accurate results about wave speed and the conditions for propagation failure by using an interface dynamics approach that circumvents the requirement of fast modulation. Next we turn our attention to forms of disorder that arise via the variation of firing rate properties across the tissue. To model this we draw parameters of the firing rate function from a distribution with prescribed spatial correlations and analyse the corresponding fluctuations in the wave speed. Finally we consider generalisations of the model to incorporate adaptation and stochastic forcing and show how recent numerical techniques developed for stochastic partial differential equations can be used to determine the wave speed by minimising the L-2 norm of a travelling disordered activity profile against a fixed test function.
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| 4. |
- Douanla Yonta, Hermann, 1982, et al.
(författare)
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Homogenization of a nonlinear elliptic problem with large nonlinear potential
- 2012
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Ingår i: Applicable Analysis. - : Informa UK Limited. - 0003-6811 .- 1563-504X. ; 91:6, s. 1205-1218
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Tidskriftsartikel (refereegranskat)abstract
- Homogenization is studied for a nonlinear elliptic boundary-value problem with a large nonlinear potential. More specifically we are interested in the asymptotic behaviour of a sequence of p-Laplacians of the form -div(a(x/epsilon)vertical bar Du(epsilon vertical bar)(p-2)Du(epsilon)) + 1/epsilon V(x/epsilon)vertical bar u(epsilon)vertical bar(p-2)u(epsilon) = f. It is shown that, under a centring condition on the potential V, there exists a two-scale homogenized system with solution (u, u(1)) such that the sequence u(epsilon) of solutions converges weakly to u in W-1,W-p and the gradients D(x)u(epsilon) two-scale converges weakly to D(x)u+D(y)u(1) in L-p, respectively. We characterize the limit system explicitly by means of two-scale convergence and a new convergence result.
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| 5. |
- Douanla Yonta, Hermann, 1982, et al.
(författare)
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Reiterated Homogenization of Linear Eigenvalue Problems in Multiscale Perforated Domains Beyond the Periodic Setting
- 2011
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Ingår i: Communications in Mathematical Analysis. - 1938-9787. ; 11:1, s. 61-93
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Tidskriftsartikel (refereegranskat)abstract
- Reiterated homogenization of linear elliptic Neuman eigenvalue problems in multiscale perforated domains is considered beyond the periodic setting. The classical periodicity hypothesis on the coefficients of the operator is here substituted on each microscale by an abstract hypothesis covering a large set of concrete behaviors such as the periodicity, the almost periodicity, the weakly almost periodicity and many more besides. Furthermore, the usual double periodicity is generalized by considering a type of structure where the perforations on each scale follow not only the periodic distribution but also more complicated but realistic ones. Our main tool is Nguetseng’s Sigma convergence.
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| 6. |
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| 7. |
- Francu, J., et al.
(författare)
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Some remarks on two-scale convergence and periodic unfolding
- 2012
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Ingår i: Applications of Mathematics. - : Institute of Mathematics, Czech Academy of Sciences. - 0862-7940 .- 1572-9109. ; 57:4, s. 359-375
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Tidskriftsartikel (refereegranskat)abstract
- The paper discusses some aspects of the adjoint definition of two-scale convergence based on periodic unfolding. As is known this approach removes problems concerning choice of the appropriate space for admissible test functions. The paper proposes a modified unfolding which conserves integral of the unfolded function and hence simplifies the proofs and its application in homogenization theory. The article provides also a self-contained introduction to two-scale convergence and gives ideas for generalization to non-periodic homogenization.
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| 9. |
- Nguetseng, Gabriel, et al.
(författare)
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Asymptotic Analysis for a Weakly Damped Wave Equation with Application to a Problem Arising in Elasticity
- 2010
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Ingår i: JOURNAL OF FUNCTION SPACES AND APPLICATIONS. - : Hindawi Limited. - 0972-6802 .- 1758-4965. ; 8:1, s. 17-52
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Tidskriftsartikel (refereegranskat)abstract
- The present work is devoted to the study of homogenization of the weakly damped wave equation∫Ωρε∂2uε∂t2(t)⋅υdx+2ε2μ∫ΩfεEij(∂uε∂t(t))Eij(υ)dx+ε2λ∫Ωfεdiv(∂uε∂t(t))div υdx+ϑ∫Ωfεdiv(uε(t))divυdx=∫Ωf(t)⋅υdx for all υ=(υ1,υ2,υ3)∈Vε(0<t<T), with initial conditionsuε(0)=∂uε∂t(0)=ω (the origin in ℝ3). Convergence homogenization results are achieved using the two-scale convergence theory.
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