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| 2. |
- Babuska, Ivo, et al.
(författare)
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A stochastic collocation method for elliptic partial differential equations with random input data
- 2007
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 45:3, s. 1005-1034
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms ( input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuska, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 ( 2004), pp. 800-825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.
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| 3. |
- Babuska, I., et al.
(författare)
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Galerkin finite element approximations of stochastic elliptic partial differential equations
- 2004
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Ingår i: SIAM Journal on Numerical Analysis. - 0036-1429 .- 1095-7170. ; 42:2, s. 800-825
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Tidskriftsartikel (refereegranskat)abstract
- We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the computations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the computation of the expected value of the solution. The first method generates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element variational formulation. The Monte Carlo method then uses these approximations to compute corresponding sample averages. The second method is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, of either the h- or p-version, then approximates the corresponding deterministic solution, yielding approximations of the desired statistics. We present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy. This comparison suggests intuitive conditions for an optimal selection of the numerical approximation.
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| 5. |
- Burman, Erik, et al.
(författare)
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Continuous interior penalty finite element method for Oseen's equations
- 2006
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 1095-7170 .- 0036-1429. ; 44:3, s. 1248 - 1274
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we present an extension of the continuous interior penalty method of Douglas and Dupont [Interior penalty procedures for elliptic and parabolic Galerkin methods, in Computing Methods in Applied Sciences, Lecture Notes in Phys. 58, Springer-Verlag, Berlin, 1976, pp. 207-216] to Oseen's equations. The method consists of a stabilized Galerkin formulation using equal order interpolation for pressure and velocity. To counter instabilities due to the pressure/velocity coupling, or due to a high local Reynolds number, we add a stabilization term giving L2-control of the jump of the gradient over element faces (edges in two dimensions) to the standard Galerkin formulation. Boundary conditions are imposed in a weak sense using a consistent penalty formulation due to Nitsche. We prove energy-type a priori error estimates independent of the local Reynolds number and give some numerical examples recovering the theoretical results.
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| 6. |
- Chung, E., et al.
(författare)
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Convergence analysis of fully discrete finite volume methods for Maxwell's equations in nonhomogeneous media
- 2005
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Ingår i: SIAM Journal on Numerical Analysis. - 0036-1429 .- 1095-7170. ; 43:1, s. 303-317
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Tidskriftsartikel (refereegranskat)abstract
- We will consider both explicit and implicit fully discrete finite volume schemes for solving three-dimensional Maxwell's equations with discontinuous physical coefficients on general polyhedral domains. Stability and convergence for both schemes are analyzed. We prove that the schemes are second order accurate in time. Both schemes are proved to be first order accurate in space for the Voronoi-Delaunay grids and second order accurate for nonuniform rectangular grids. We also derive explicit expressions for the dependence on the physical parameters in all estimates.
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| 7. |
- Chung, Eric T., et al.
(författare)
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Optimal Discontinuous Galerkin Methods for the Acoustic Wave Equation in Higher Dimensions
- 2009
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial and Applied Mathematics. - 0036-1429 .- 1095-7170. ; 47:5, s. 3820-3848
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we developed and analyzed a new class of discontinuous Galerkin (DG) methods for the acoustic wave equation in mixed form. Traditional mixed finite element (FE) methods produce energy conserving schemes, but these schemes are implicit, making the time-stepping inefficient. Standard DG methods give explicit schemes, but these approaches are typically dissipative or suboptimally convergent, depending on the choice of numerical fluxes. Our new method can be seen as a compromise between these two kinds of techniques, in the way that it is both explicit and energy conserving, locally and globally. Moreover, it can be seen as a generalized version of the Raviart-Thomas FE method and the finite volume method. Stability and convergence of the new method are rigorously analyzed, and we have shown that the method is optimally convergent. Furthermore, in order to apply the new method for unbounded domains, we proposed a new way to handle the second order absorbing boundary condition. The stability of the resulting numerical scheme is analyzed.
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| 8. |
- Chung, Eric T., et al.
(författare)
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Optimal discontinuous Galerkin methods for wave propagation
- 2006
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 44:5, s. 2131-2158
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Tidskriftsartikel (refereegranskat)abstract
- We have developed and analyzed a new class of discontinuous Galerkin methods (DG) which can be seen as a compromise between standard DG and the finite element (FE) method in the way that it is explicit like standard DG and energy conserving like FE. In the literature there are many methods that achieve some of the goals of explicit time marching, unstructured grid, energy conservation, and optimal higher order accuracy, but as far as we know only our new algorithms satisfy all the conditions. We propose a new stability requirement for our DG. The stability analysis is based on the careful selection of the two FE spaces which verify the new stability condition. The convergence rate is optimal with respect to the order of the polynomials in the FE spaces. Moreover, the convergence is described by a series of numerical experiments.
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| 9. |
- Gaidashev, Denis, et al.
(författare)
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On numerical algorithms for the solution of a Beltrami equation
- 2008
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 46:5, s. 2238-2253
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Tidskriftsartikel (refereegranskat)abstract
- The paper concerns numerical algorithms for solving the Beltrami equation f (z) over bar (z) = mu( z) fz( z) for a compactly supported mu. First, we study an e. cient algorithm that has been proposed in [ P. Daripa, J. Comput. Phys., 106 ( 1993), pp. 355 - 365] and [ P. Daripa and D. Mashat, Numer. Algorithms, 18 ( 1998), pp. 133 - 157] and present its rigorous justi. cation. We then propose a different scheme for solving the Beltrami equation which has a comparable speed and accuracy, but has the virtue of easier implementation by avoiding the use of the Hilbert transform. The present paper can also be viewed as a prologue to one important application of the Beltrami equation: it provides a detailed description of the algorithm that has been used in [ D. Gaidashev, Nonlinearity, 20 ( 1998), pp. 713 - 741] and [ D. Gaidashev and M. Yampolsky, Experiment. Math., 16 ( 2007), pp. 215 - 226] to address an important issue in complex dynamics - conjectural universality for Siegel disks.
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| 10. |
- Grzhibovskis, Richards, 1978, et al.
(författare)
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A convolution-thresholding approximation of generalized curvature flows
- 2005
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Ingår i: SIAM Journal on Numerical Analysis. - 0036-1429 .- 1095-7170. ; 42:6, s. 2652-2670
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Tidskriftsartikel (refereegranskat)abstract
- We construct a convolution-thresholding approximation scheme for the geometric surface evolution in the case when the velocity of the surface at each point is a given function of the mean curvature. Conditions for the monotonicity of the scheme are found and the convergence of the approximations to the corresponding viscosity solution is proved. We also discuss some aspects of the numerical implementation of such schemes and present several numerical results.
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