We consider the augmented mixed finite element method proposed in Barrios et al. (Comput Methods Appl Mech Eng 283:909-922, 2015) for Darcy flow. We develop the a priori and a posteriori error analyses taking into acc...
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ISBN:
(纸本)9783319257273;9783319257259
We consider the augmented mixed finite element method proposed in Barrios et al. (Comput Methods Appl Mech Eng 283:909-922, 2015) for Darcy flow. We develop the a priori and a posteriori error analyses taking into account the approximation of the Neumann boundary condition. We derive an a posteriori error indicator that consists of two residual terms on interior elements and an additional term that accounts for the error in the boundary condition on boundary elements. We prove that the error indicator is reliable and locally efficient on interior elements. Numerical experiments illustrate the good performance of the adaptive algorithm.
We consider the solution of large linear systems of equations that arise when two-dimensional singularly perturbed reaction-diffusion equations are discretized. Standard methods for these problems, such as central fin...
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ISBN:
(纸本)9783319257273;9783319257259
We consider the solution of large linear systems of equations that arise when two-dimensional singularly perturbed reaction-diffusion equations are discretized. Standard methods for these problems, such as central finite differences, lead to system matrices that are positive definite. The direct solvers of choice for such systems are based on Cholesky factorisation. However, as observed in MacLachlan and Madden (SIAM J Sci Comput 35:A2225-A2254, 2013), these solvers may exhibit poor performance for singularly perturbed problems. We provide an analysis of the distribution of entries in the factors based on their magnitude that explains this phenomenon, and give bounds on the ranges of the perturbation and discretization parameters where poor performance is to be expected.
In the description of water waves, dispersion is one of the most important physical properties;it specifies the propagation speed as function of the wavelength. Accurate modelling of dispersion is essential to obtain ...
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ISBN:
(纸本)9783319198002;9783319197999
In the description of water waves, dispersion is one of the most important physical properties;it specifies the propagation speed as function of the wavelength. Accurate modelling of dispersion is essential to obtain high-quality wave propagation results. The relation between speed and wavelength is given by a non-algebraic relation;for finite element/difference methods this relation has to be approximated and leads to restrictions for waves that are propagated correctly. By using a spectral implementation dispersion can be dealt with exactly above flat bottom using a pseudo-differential operator so that all wavelengths can be propagated correctly. However, spectral methods are most commonly applied for problems in simple domains, while most water wave applications need complex geometries such as (harbour) walls, varying bathymetry, etc.;also breaking of waves requires a local procedure at the unknown position of breaking. This paper deals with such inhomogeneities in space;the models are formulated using Fourier integral operators and include non-trivial localization methods. The efficiency and accuracy of a so-called spatial-spectral implementation is illustrated here for a few test cases: wave run-up on a coast, wave reflection at a wall and the breaking of a focussing wave. These methods are included in HAWASSI software (Hamiltonian Wave-Ship-Structure Interaction) that has been developed over the past years.
We present two-sided a posteriori error estimates for isogeometric discretization of elliptic problems. These estimates, derived on functional grounds, provide robust, guaranteed and sharp two-sided bounds of the exac...
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ISBN:
(数字)9783319233154
ISBN:
(纸本)9783319233154;9783319233147
We present two-sided a posteriori error estimates for isogeometric discretization of elliptic problems. These estimates, derived on functional grounds, provide robust, guaranteed and sharp two-sided bounds of the exact error in the energy norm. Moreover, since these estimates do not contain any unknown/generic constants, they are fully computable, and thus provide quantitative information on the error. The numerical realization and the quality of the computed error distribution are addressed. The potential of the proposed estimates are illustrated using several computational examples.
We present a locally conservative spectral least-squares formulation for the scalar diffusion-reaction equation in curvilinear coordinates. Careful selection of a least squares functional and compatible finite dimensi...
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ISBN:
(纸本)9783319198002;9783319197999
We present a locally conservative spectral least-squares formulation for the scalar diffusion-reaction equation in curvilinear coordinates. Careful selection of a least squares functional and compatible finite dimensional subspaces for the solution space yields the conservation properties. Numerical examples confirm the theoretical properties of the method.
We present an algorithm that reformulates existing methods to construct higher-order mimetic differential operators. Constrained linear optimization is the key idea of this resulting algorithm. The authors exemplified...
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ISBN:
(纸本)9783319198002;9783319197999
We present an algorithm that reformulates existing methods to construct higher-order mimetic differential operators. Constrained linear optimization is the key idea of this resulting algorithm. The authors exemplified this algorithm by constructing an eight-order-accurate one-dimensional mimetic divergence operator. The algorithm computes the weights that impose the mimetic condition on the constructed operator. However, for higher orders, the computation of valid weights can only be achieved through this new algorithm. Specifically, we provide insights on the computational implementation of the proposed algorithm, and some results of its application in different test cases. Results show that for all of the proposed test cases, the proposed algorithm effectively solves the problem of computing valid weights, thus constructing higher-order mimetic operators.
Strong stability preserving (SSP) high order time discretizations were developed to address the need for nonlinear stability properties in the numerical solution of hyperbolic partial differential equations with disco...
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ISBN:
(纸本)9783319198002;9783319197999
Strong stability preserving (SSP) high order time discretizations were developed to address the need for nonlinear stability properties in the numerical solution of hyperbolic partial differential equations with discontinuous solutions. These methods preserve the monotonicity properties (in any norm, seminorm or convex functional) of the spatial discretization coupled with first order Euler time stepping. This review paper describes the state of the art in SSP methods.
We outline and extend results for an explicit local time stepping (LTS) strategy designed to operate with the discontinuous Galerkin spectral element method (DGSEM). The LTS procedure is derived from Adams-Bashforth m...
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ISBN:
(纸本)9783319198002;9783319197999
We outline and extend results for an explicit local time stepping (LTS) strategy designed to operate with the discontinuous Galerkin spectral element method (DGSEM). The LTS procedure is derived from Adams-Bashforth multirate time integration methods. The new results of the LTS method focus on parallelization and reformulation of the LTS integrator to maintain conservation. Discussion is focused on a moving mesh implementation, but the procedures remain applicable to static meshes. In numerical tests, we demonstrate the strong scaling of a parallel, LTS implementation and compare the scaling properties to a parallel, global time stepping (GTS) Runge-Kutta implementation. We also present time-step refinement studies to show that the redesigned, conservative LTS approximations are spectrally accurate in space and have design temporal accuracy.
In this contribution we present a sub-cell discretization method for the computation of the interface velocities involved in the convective terms of the incompressible Navier-Stokes equations. We compute an interface ...
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ISBN:
(纸本)9783319198002;9783319197999
In this contribution we present a sub-cell discretization method for the computation of the interface velocities involved in the convective terms of the incompressible Navier-Stokes equations. We compute an interface velocity by solving a local two-point boundary value problem (BVP) iteratively. To account for the two-dimensionality of the interface velocity we introduce a constant cross-flux term in our computation. The discretization scheme is used to simulate the flow in a lid-driven cavity.
In this paper, a boundary value problem for a system of two singularly perturbed second order delay differential equations is considered on the interval [0, 2]. The components of the solution of this system exhibit bo...
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ISBN:
(纸本)9783319257273;9783319257259
In this paper, a boundary value problem for a system of two singularly perturbed second order delay differential equations is considered on the interval [0, 2]. The components of the solution of this system exhibit boundary layers at x = 0 and x = 2 and interior layers at x D 1. A numerical method composed of a classical finite difference scheme applied on a piecewise uniform Shishkin mesh is suggested to solve the problem. The method is proved to be first order convergent in the maximum norm uniformly in the perturbation parameters. Numerical illustration provided support the theory.
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