| 1. |
- Boon, Wietse M., et al.
(författare)
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Flux-mortar mixed finite element methods on nonmatching grids
- 2022
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 60:3, s. 1193-1225
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Tidskriftsartikel (refereegranskat)abstract
- We investigate a mortar technique for mixed finite element approximations of a class of domain decomposition saddle point problems on nonmatching grids in which the variable associated with the essential boundary condition, referred to as flux, is chosen as the coupling variable. It plays the role of a Lagrange multiplier to impose weakly continuity of the variable associated with the natural boundary condition. The flux-mortar variable is incorporated with the use of a discrete extension operator. We present well-posedness and error analysis in an abstract setting under a set of suitable assumptions, followed by a nonoverlapping domain decomposition algorithm that reduces the global problem to a positive definite interface problem. The abstract theory is illustrated for Darcy flow, where the normal flux is the mortar variable used to impose continuity of pressure, and for Stokes flow, where the velocity vector is the mortar variable used to impose continuity of normal stress. In both examples, suitable discrete extension operators are developed and the assumptions from the abstract theory are verified. Numerical studies illustrating the theoretical results are presented for Darcy flow.
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| 2. |
- Burman, E., et al.
(författare)
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CUT FINITE ELEMENT METHOD FOR DIVERGENCE-FREE APPROXIMATION OF INCOMPRESSIBLE FLOW : A LAGRANGE MULTIPLIER APPROACH
- 2024
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial and Applied Mathematics. - 0036-1429 .- 1095-7170. ; 62:2, s. 893-918
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Tidskriftsartikel (refereegranskat)abstract
- In this note, we design a cut finite element method for a low order divergence-free element applied to a boundary value problem subject to Stokes' equations. For the imposition of Dirichlet boundary conditions, we consider either Nitsche's method or a stabilized Lagrange multiplier method. In both cases, the normal component of the velocity is constrained using a multiplier, different from the standard pressure approximation. The divergence of the approximate velocities is pointwise zero over the whole mesh domain, and we derive optimal error estimates for the velocity and pressures, where the error constant is independent of how the physical domain intersects the computational mesh, and of the regularity of the pressure multiplier imposing the divergence-free condition.
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| 3. |
- Duru, Kenneth, et al.
(författare)
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A Conservative and Energy Stable Ddiscontinuous Spectral Element Method for The Shifted WaveEquation in Second Order Form
- 2022
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Ingår i: SIAM Journal on Numerical Analysis. - Philadephia : Siam Publications. - 0036-1429 .- 1095-7170. ; 60:4, s. 1631-1664
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we develop a provably energy stable and conservative discontinuous spectral element method for the shifted wave equation in second order form. The proposed method combines the advantages and central ideas of the following very successful numerical techniques: the summation-by-parts finite difference method, the spectral method, and the discontinuous Galerkin method. We prove energy stability and the discrete conservation principle and derive error estimates in the energy norm for the (1+1)-dimensions shifted wave equation in second order form. The energy-stability results, discrete conservation principle, and the error estimates generalize to multiple dimensions using tensor products of quadrilateral and hexahedral elements. Numerical experiments, in (1+1)-dimensions and (2+1)-dimensions, verify the theoretical results and demonstrate optimal convergence of L2 numerical errors at subsonic, sonic and supersonic regimes.
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| 4. |
- Eriksson, Sofia, et al.
(författare)
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Summation-by-Parts Approximations of the Second Derivative : Pseudoinverse and Revisitation of a High Order Accurate Operator
- 2021
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Ingår i: SIAM Journal on Numerical Analysis. - : SIAM Publications. - 0036-1429 .- 1095-7170. ; 59:5, s. 2669-2697
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Tidskriftsartikel (refereegranskat)abstract
- We consider finite difference approximations of the second derivative, exemplified in Poisson's equation, the heat equation, and the wave equation. The finite difference operators satisfy a summation-by-parts (SBP) property, which mimics the integration-by-parts principle. Since the operators approximate the second derivative, they are singular by construction. When imposing boundary conditions weakly, these operators are modified using simultaneous approximation terms. The modification makes the discretization matrix nonsingular for most choices of boundary conditions. Recently, inverses of such matrices were derived. However, for problems with only Neumann boundary conditions, the modified matrices are still singular. For such matrices, we have derived an explicit expression for the Moore-Penrose inverse, which can be used for solving elliptic problems and some time-dependent problems. For this explicit expression to be valid, it is required that the modified matrix does not have more than one zero eigenvalue. This condition holds for the SBP operators with second and fourth order accurate interior stencil. For the sixth order accurate case, we have reconstructed the operator with a free parameter and show that there can be more than one zero eigenvalue. We have performed a detailed analysis on the free parameter to improve the properties of the second derivative SBP operator. We complement the derivations by numerical experiments to demonstrate the improvements.
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| 5. |
- Glaubitz, Jan, et al.
(författare)
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Summation-by-Parts Operators for General Function Spaces
- 2023
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial and Applied Mathematics. - 0036-1429 .- 1095-7170. ; 61:2, s. 733-754
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Tidskriftsartikel (refereegranskat)abstract
- Summation-by-parts (SBP) operators are popular building blocks for systematically developing stable and high-order accurate numerical methods for time-dependent differential equations. The main idea behind existing SBP operators is that the solution is assumed to be well approximated by polynomials up to a certain degree, and the SBP operator should therefore be exact for them. However, polynomials might not provide the best approximation for some problems, and other approximation spaces may be more appropriate. In this paper, a theory for SBP operators based on general function spaces is developed. We demonstrate that most of the established results for polynomial-based SBP operators carry over to this general class of SBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently known. We exemplify the general theory by considering trigonometric, exponential, and radial basis functions.
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| 6. |
- Henning, Patrick, 1983-, et al.
(författare)
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Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem : Global convergence and computational efficiency
- 2020
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 58:3, s. 1744-1772
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Tidskriftsartikel (refereegranskat)abstract
- We propose a new normalized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem based on an energy inner product that depends on time through the density of the flow itself. The gradient flow is well-defined and converges to an eigenfunction. For ground states we can quantify the convergence speed as exponentially fast where the rate depends on spectral gaps of a linearized operator. The forward Euler time discretization of the flow yields a numerical method which generalizes the inverse iteration for the nonlinear eigenvalue problem. For sufficiently small time steps, the method reduces the energy in every step and converges globally in H I- to an eigenfunction. In particular, for any nonnegative starting value, the ground state is obtained. A series of numerical experiments demonstrates the computational efficiency of the method and its competitiveness with established discretizations arising from other gradient flows for this problem.
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| 7. |
- Kovacs, Mihaly, 1977, et al.
(författare)
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Mittag-leffler euler integrator for a stochastic fractional order equation with additive noise
- 2020
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Ingår i: SIAM Journal on Numerical Analysis. - 0036-1429 .- 1095-7170. ; 58:1, s. 66-85
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Tidskriftsartikel (refereegranskat)abstract
- Motivated by fractional derivative models in viscoelasticity, a class of semilinear stochastic Volterra integro-differential equations, and their deterministic counterparts, are considered. A generalized exponential Euler method, named here the Mittag-Leffler Euler integrator, is used for the temporal discretization, while the spatial discretization is performed by the spectral Galerkin method. The temporal rate of strong convergence is found to be (almost) twice compared to when the backward Euler method is used together with a convolution quadrature for time discretization. Numerical experiments that validate the theory are presented. © 2020 Society for Industrial and Applied Mathematics.
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| 8. |
- Kovács, Mihály, et al.
(författare)
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Mittag-Leffler Euler integrator for a stochastic fractional order equation with additive noise
- 2020
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 58:1, s. 66-85
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Tidskriftsartikel (refereegranskat)abstract
- Motivated by fractional derivative models in viscoelasticity, a class of semilinear stochastic Volterra integro-differential equations, and their deterministic counterparts, are considered. A generalized exponential Euler method, named here the Mittag--Leffler Euler integrator, is used for the temporal discretization, while the spatial discretization is performed by the spectral Galerkin method. The temporal rate of strong convergence is found to be (almost) twice compared to when the backward Euler method is used together with a convolution quadrature for time discretization. Numerical experiments that validate the theory are presented.
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| 9. |
- Lundquist, Tomas, 1986-, et al.
(författare)
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A Method-of-Lines Framework for Energy Stable Arbitrary Lagrangian–Eulerian Methods
- 2023
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial and Applied Mathematics. - 0036-1429 .- 1095-7170. ; 61:5, s. 2327-2351
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Tidskriftsartikel (refereegranskat)abstract
- We present a novel framework based on semi-bounded spatial operators for analyzing and discretizing initial boundary value problems on moving and deforming domains. This development extends an existing framework for well-posed problems and energy stable discretizations from stationary domains to the general case, including arbitrary mesh motion. In particular, we show that an energy estimate derived in the physical coordinate system is equivalent to a semi-bounded property with respect to a stationary reference domain. The continuous analysis leading up to this result is based on a skew-symmetric splitting of the material time derivative and thus relies on the property of integration-by-parts. Following this, a mimetic energy stable arbitrary Lagrangian–Eulerian framework for semi-discretization is formulated, based on approximating the material time derivative in a way consistent with discrete summation-by-parts. Thanks to the semi-bounded property, a method-of-lines approach using standard explicit or implicit time integration schemes can be applied to march the system forward in time. The same type of stability arguments as for the corresponding stationary domain problem applies, without regard to additional properties such as discrete geometric conservation. As an additional bonus we demonstrate that discrete geometric conservation, in the sense of exact free-stream preservation, can still be achieved in an automatic way with the new framework. However, we stress that this is not necessary for stability.
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| 10. |
- Maier, Roland, 1993
(författare)
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A high-order approach to elliptic multiscale problems with general unstructured coefficients
- 2021
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Ingår i: SIAM Journal on Numerical Analysis. - 1095-7170 .- 0036-1429. ; 59:2, s. 1067-1089
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Tidskriftsartikel (refereegranskat)abstract
- We propose a multiscale approach for an elliptic multiscale setting with general unstructured diffusion coefficients that is able to achieve high-order convergence rates with respect to the mesh parameter and the polynomial degree. The method allows for suitable localization and does not rely on additional regularity assumptions on the domain, the diffusion coefficient, or the exact (weak) solution as typically required for high-order approaches. Rigorous a priori error estimates are presented with respect to the involved discretization parameters, and the interplay between these parameters as well as the performance of the method are studied numerically.
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