| 1. |
- Efraimsson, Gunilla
(författare)
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A numerical method for the first-order wave equation with discontinuous initial data
- 1998
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Ingår i: Numerical Methods for Partial Differential Equations. - : John Wiley & Sons. - 0749-159X .- 1098-2426. ; 138:3, s. 353-365
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a method, constructed such that numerical solutions of the wave equation are well behaved when the solutions also contain discontinuities. The wave equation serves as a model problem for the Euler equations when the solution contains a contact discontinuity. Numerical computations of linear equations and the Euler equations in one and two dimensions are presented
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| 2. |
- Hansbo, Peter
(författare)
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Moving finite element methods by use of space-time elements : I. Scalar problems
- 1998
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Ingår i: Numerical Methods for Partial Differential Equations. - 0749-159X .- 1098-2426. ; 14:2, s. 251-262
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Tidskriftsartikel (refereegranskat)abstract
- This article deals with moving finite element methods by use of the time-discontinuous Galerkin formulation in combination with oriented space-time meshes. A principle for mesh orientation in space-time based on minimization of the residual, related to adaptive error control via an a posteriori error estimate, is presented. The relation to Miller's moving finite element method is discussed. The article deals with scalar problems: systems will be treated in a companion article.
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| 3. |
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| 4. |
- Asadzadeh, Mohammad, 1952, et al.
(författare)
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Discontinuous Galerkin and multiscale variational schemes for a coupled damped nonlinear system of Schrödinger equations
- 2013
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Ingår i: Numerical Methods for Partial Differential Equations. - : Wiley. - 0749-159X .- 1098-2426. ; 29:6, s. 1912-1945
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Tidskriftsartikel (refereegranskat)abstract
- In this article, we study a streamline diffusion-based discontinuous Galerkin approximation for the numerical solution of a coupled nonlinear system of Schrödinger equations and extend the resulting method to a multiscale variational scheme. We prove stability estimates and derive optimal convergence rates due to the maximal available regularity of the exact solution. In the weak formulation, to make the underlying bilinear form coercive, it was necessary to supply the equation system with an artificial viscosity term with a small coefficient of order proportional to a power of mesh size. We justify the theory by implementing an example of an application of the time-dependent Schrödinger equation in the coupled ultrafast laser.
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| 5. |
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| 6. |
- Babuska, I., et al.
(författare)
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Reliability of computational science
- 2007
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Ingår i: Numerical Methods for Partial Differential Equations. - : Wiley. - 0749-159X .- 1098-2426. ; 23:4, s. 753-784
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Tidskriftsartikel (refereegranskat)abstract
- Today's computers allow us to simulate large, complex physical problems. Many times the mathematical models describing such problems are based on a relatively small amount of available information such as experimental measurements. The question arises whether the computed data could be used as the basis for decision in critical engineering, economic, and medicine applications. The representative list of engineering accidents occurred in the past years and their reasons illustrate the question. The paper describes a general framework for verification and validation (V&V) which deals with this question. The framework is then applied to an illustrative engineering problem, in which the basis for decision is a specific quantity of interest, namely the probability that the quantity does not exceed a given value. The V&V framework is applied and explained in detail. The result of the analysis is the computation of the failure probability as well as a quantification of the confidence in the computation, depending on the amount of available experimental data.
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| 7. |
- Bazarganzadeh, Mahmoudreza, et al.
(författare)
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Numerical Approximation of One Phase Quadrature Domains
- 2013
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Ingår i: Numerical Methods for Partial Differential Equations. - : Wiley. - 0749-159X .- 1098-2426. ; 29:5, s. 1709-1728
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- In this work, we present two numerical schemes for a free boundary problem called one phase quadrature domain. In the first method by applying the proprieties of given free boundary problem, we derive a method that leads to a fast iterative solver. The iteration procedure is adapted in order to work in the case when topology changes. The second method is based on shape reconstruction to establish an efficient Shape-Quasi-Newton-Method. Various numerical experiments confirm the efficiency of the derived numerical methods.
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| 8. |
- Burman, Erik, et al.
(författare)
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Stabilized Crouzeix-Raviart element for the Darcy-Stokes problem
- 2005
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Ingår i: Numerical Methods for Partial Differential Equations. - : Wiley. - 0749-159X .- 1098-2426. ; 21:5, s. 986-997
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Tidskriftsartikel (refereegranskat)abstract
- We stabilize the nonconforming Crouzeix-Raviart element for the Darcy-Stokes problem with terms motivated by a discontinuous Galerkin approach. Convergence of the method is shown, also in the limit of vanishing viscosity. Finally, some numerical examples verifying the theoretical predictions are presented.
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| 9. |
- Chatzipantelidis, P, et al.
(författare)
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Error estimates for a finite volume element method for parabolic equations in convex polygonal domains
- 2004
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Ingår i: Numer. Methods Partial Differential Equations. - : Wiley. - 0749-159X .- 1098-2426. ; 20:5, s. 650-674
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Tidskriftsartikel (refereegranskat)abstract
- We analyze the spatially semidiscrete piecewise linear finite volume element method for parabolic equations in a convex polygonal domain in the plane. Our approach is based on the properties of the standard finite element Ritz projection and also of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Because the domain is polygonal, special attention has to be paid to the limited regularity of the exact solution. We give sufficient conditions in terms of data that yield optimal order error estimates in L2 and H[1]. The convergence rate in the L norm is suboptimal, the same as in the corresponding finite element method, and almost optimal away from the corners. We also briefly consider the lumped mass modification and the backward Euler fully discrete method.
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| 10. |
- Chatzipantelidis, P, et al.
(författare)
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Parabolic finite volume element equations in nonconvex polygonal domains
- 2009
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Ingår i: Numerical Methods for Partial Differential Equations. - : Wiley. - 1098-2426 .- 0749-159X. ; 25:3, s. 507-525
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Tidskriftsartikel (refereegranskat)abstract
- We study spatially semidiscrete and fully discrete finite volume element approximations of the heat equation with homogeneous Dirichlet boundary conditions in a plane polygonal domain with one reentrant corner. We show that, as a result of the singularity in the solution near the reentrant corner, the convergence rate is reduced from optimal second order, similarly to what was shown for the finite element method in the earlier work []. Optimal order convergence may be restored by mesh refinement near the corners of the domain. © 2008 Wile y Periodicals, Inc.
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