| 1. |
- Aceto, Lidia, et al.
(författare)
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Fractional Laplace operator in two dimensions, approximating matrices, and related spectral analysis
- 2020
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Ingår i: Calcolo. - : Springer Science and Business Media LLC. - 0008-0624 .- 1126-5434. ; 57
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Tidskriftsartikel (refereegranskat)abstract
- In this work we review some proposals to define the fractional Laplace operator in two or more spatial variables and we provide their approximations using finite differences or the so-called Matrix Transfer Technique. We study the structure of the resulting large matrices from the spectral viewpoint. In particular, by considering the matrix-sequences involved, we analyze the extreme eigenvalues, we give estimates on conditioning, and we study the spectral distribution in the Weyl sense using the tools of the theory of Generalized Locally Toeplitz matrix-sequences. Furthermore, we give a concise description of the spectral properties when non-constant coefficients come into play. Several numerical experiments are reported and critically discussed.
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| 3. |
- Cohen, David, et al.
(författare)
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Numerical approximation and simulation of the stochastic wave equation on the sphere
- 2022
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Ingår i: Calcolo. - : Springer Science and Business Media LLC. - 0008-0624 .- 1126-5434. ; 59:3
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Tidskriftsartikel (refereegranskat)abstract
- Solutions to the stochastic wave equation on the unit sphere are approximated by spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided and shown to depend only on the smoothness of the driving noise and the initial conditions. Numerical experiments confirm the theoretical rates. The developed numerical method is extended to stochastic wave equations on higher-dimensional spheres and to the free stochastic Schrodinger equation on the unit sphere.
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| 6. |
- Dravins, Ivo, 1991-, et al.
(författare)
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Preconditioning of discrete state- and control-constrained optimal control convection-diffusion problems
- 2023
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Ingår i: Calcolo. - : Springer Nature. - 0008-0624 .- 1126-5434. ; 60
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Tidskriftsartikel (refereegranskat)abstract
- We consider the iterative solution of algebraic systems, arising in optimal control problems constrained by a partial differential equation with additional box constraints on the state and the control variables, and sparsity imposed on the control. A nonsymmetric two-by-two block preconditioner is analysed and tested for a wide range of problem, regularization and discretization parameters. The constraint equation characterizes convection-diffusion processes.
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| 7. |
- Hansbo, Anita, 1960-
(författare)
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Nonsmooth data error estimates for damped single step methods for parabolic equations in Banach space
- 1999
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Ingår i: Calcolo. - : Springer Science and Business Media LLC. - 0008-0624 .- 1126-5434. ; 36:2, s. 75-101
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Tidskriftsartikel (refereegranskat)abstract
- We consider a time discretization method for a parabolic initial boundary value problem obtained from a combination of an A-stable single step method of order p and a lower order method with good smoothing properties. Such methods, including the Crank-Nicolson method combined with the backward Euler method, were analyzed in Hilbert space by Luskin and Rannacher, and nonsmooth data error estimates of order p were obtained. We extend this result to Banach space, and also consider approximations of the time derivative. Further, we apply the results to obtain error estimates in the supremum norm for fully discrete methods obtained by discretizing the space variable by a finite element method. © Springer-Verlag 1999.
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| 8. |
- Hansbo, Peter F G, 1959, et al.
(författare)
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A discontinuous Galerkin method for the plate equation
- 2002
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Ingår i: Calcolo. - : Springer Science and Business Media LLC. - 0008-0624 .- 1126-5434. ; 39:1, s. 41-59
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Tidskriftsartikel (refereegranskat)abstract
- We present a discontinuous Galerkin method for the plate problem. The method employs a discontinuous approximation space allowing nonmatching grids and different types of approximation spaces. Continuity is enforced weakly through the variational form. Discrete approximations of the normal and twisting moments and the transversal force, which satisfy the equilibrium condition on an element level, occur naturally in the method. We show optimal a priori error estimates in various norms and investigate locking phenomena when certain stabilization parameters tend to infinity. Finally, we relate the method to two classical elements: the nonconforming Morley element and the C^1 Argyris element.
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| 10. |
- Mele, Giampaolo, 1989-
(författare)
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The infinite Lanczos method for symmetric nonlinear eigenvalue problems
- 2023
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Ingår i: Calcolo. - : Springer Nature. - 0008-0624 .- 1126-5434. ; 60:2
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Tidskriftsartikel (refereegranskat)abstract
- A new iterative method for solving large scale symmetric nonlinear eigenvalue problems is presented. We firstly derive an infinite dimensional symmetric linearization of the nonlinear eigenvalue problem, then we apply the indefinite Lanczos method to this specific linearization, resulting in a short-term recurrence. We show how, under specific assumption on the starting vector, this method can be carried out in finite arithmetic and how the exploitation of the problem structure leads to improvements in terms of computation time. The eigenpair approximations are extracted with the nonlinear Rayleigh-Ritz procedure combined with a specific choice of the projection space. We illustrate how this extraction technique resolves the instability issues that may occur due to the loss of orthogonality in many standard Lanczos-type methods.
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