Application of spectral element method for solving Sobolev equations with error estimation
TL;DR: In this article, a semi-discrete solution of the Sobolev equations is proposed by discretizing the spatial derivatives via the Legendre spectral element method (LSEM) and the Lagrange polynomial based on the Gauss-Legendre-Lobatto (GLL) points.
About: This article is published in Applied Numerical Mathematics.The article was published on 2020-12-01. It has received 15 citations till now. The article focuses on the topics: Spectral element method & Sobolev space.
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TL;DR: In this paper, a numerical approach for finding the approximate solution of the Sobolev model is presented. But the major disadvantage of global techniques is the high computational burden of solving large linear systems.
19 citations
01 Jan 1999
TL;DR: In this article, a spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two-and three-dimensional geometries.
Abstract: A spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an efficient tool for simulating elastic wave propagation in realistic geological structures in two- and three-dimensional geometries. The computational domain is discretized into quadrangles, or hexahedra, defined with respect to a reference unit domain by an invertible local mapping. Inside each reference element, the numerical integration is based on the tensor-product of a Gauss–Lobatto–Legendre 1-D quadrature and the solution is expanded onto a discrete polynomial basis using Lagrange interpolants. As a result, the mass matrix is always diagonal, which drastically reduces the computational cost and allows an efficient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor/multicorrector format. Long term energy conservation and stability properties are illustrated as well as the efficiency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two-dimensional problems obtained by other methods. The potentiality of the method is then illustrated by studying a simple three-dimensional model. Very accurate modelling of Rayleigh wave propagation and surface diffraction is obtained at a low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three-dimensional surface topographies. Copyright © 1999 John Wiley & Sons, Ltd.
18 citations
TL;DR: An efficient finite difference/generalized Hermite spectral method for the distributed-order time-fractional reaction-diffusion equation on one-, two-, and three-dimensional unbounded domains and the optimal error estimate is derived for the space approximation.
Abstract: Distributed-order fractional differential equations, where the differential order is distributed over a range of values rather than being just a fixed value as it is in the classical differential equations, offer a powerful tool to describe multi-physics phenomena. In this article, we develop and analyze an efficient finite difference/generalized Hermite spectral method for the distributed-order time-fractional reaction-diffusion equation on one-, two-, and three-dimensional unbounded domains. Considering the Gauss-Legendre quadrature rule for the distributed integral term in temporal direction, we first approximate the original distributed-order time-fractional problem by the multi-term time-fractional differential equation. Then, we apply the L2- 1 σ formula for the discretization of the multi-term Caputo fractional derivatives. Moreover, we employ the generalized Hermite functions with scaling factor for the spectral approximation in space. The detailed implementations of the method are presented for one-, two-, and three-dimensional cases of the fractional problem. The stability and convergence of the method are strictly established, which shows that the proposed method is unconditionally stable and convergent with second-order accuracy in time. In addition, the optimal error estimate is derived for the space approximation. Finally, we perform numerical examples to support the theoretical claims.
10 citations
TL;DR: In this paper, the authors apply the Heydari-Hosseininia nonsingular fractional derivative for defining a variable-order fractional version of the Sobolev equation.
Abstract: This paper applies the Heydari–Hosseininia nonsingular fractional derivative for defining a variable-order fractional version of the Sobolev equation. The orthonormal shifted discrete Legendre polynomials, as an appropriate family of basis functions, are employed to generate an operational matrix method for this equation. A new fractional operational matrix related to these polynomials is extracted and employed to construct the presented method. Using this approach, an algebraic system of equations is obtained instead of the original variable-order equation. The numerical solution of this system can be found easily. Some numerical examples are provided for verifying the accuracy of the generated approach.
6 citations
TL;DR: In this article , two classes of one-parameter orthogonal spline collocation (OSC) methods are constructed for solving initial boundary value problems with time-variable delay.
5 citations
References
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Book•
07 Sep 2011
TL;DR: In this article, the theory of conjugate convex functions is introduced, and the Hahn-Banach Theorem and the closed graph theorem are discussed, as well as the variations of boundary value problems in one dimension.
Abstract: Preface.- 1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions.- 2. The Uniform Boundedness Principle and the Closed Graph Theorem. Unbounded Operators. Adjoint. Characterization of Surjective Operators.- 3. Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity.- 4. L^p Spaces.- 5. Hilbert Spaces.- 6. Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators.- 7. The Hille-Yosida Theorem.- 8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension.- 9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions.- 10. Evolution Problems: The Heat Equation and the Wave Equation.- 11. Some Complements.- Problems.- Solutions of Some Exercises and Problems.- Bibliography.- Index.
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TL;DR: In this article, the authors provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation.
Abstract: This is the softcover reprint of the very popular hardcover edition. This book deals with the numerical approximation of partial differential equations. Its scope is to provide a thorough illustration of numerical methods, carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is one of its main features. Many kinds of problems are addressed. A comprehensive theory of Galerkin method and its variants, as well as that of collocation methods, are developed for the spatial discretization. These theories are then specified to two numerical subspace realizations of remarkable interest: the finite element method and the spectral method.
2,383 citations
TL;DR: In this article, a spectral element method was proposed for numerical solution of the Navier-Stokes equations, where the computational domain is broken into a series of elements, and the velocity in each element is represented as a highorder Lagrangian interpolant through Chebyshev collocation points.
2,133 citations
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04 Apr 2006
TL;DR: In this article, the authors have incorporated into this new edition the many improvements in the algorithms and the theory of spectral methods that have been made since then, and the discussion of direct and iterative solution methods is also greatly expanded.
Abstract: Since the publication of ''Spectral Methods in Fluid Dynamics'' 1988, spectral methods have become firmly established as a mainstream tool for scientific and engineering computation. The authors of that book have incorporated into this new edition the many improvements in the algorithms and the theory of spectral methods that have been made since then. This latest book retains the tight integration between the theoretical and practical aspects of spectral methods, and the chapters are enhanced with material on the Galerkin with numerical integration version of spectral methods. The discussion of direct and iterative solution methods is also greatly expanded. (orig.)
1,805 citations