Iterative Solution of Shifted Positive-Definite Linear Systems Arising in a Numerical Method for the Heat Equation Based on Laplace Transformation and Quadrature

McLean, William; Thomée, Vidar

doi:10.1017/s1446181112000107

Cited by 2 publications

(4 citation statements)

References 17 publications

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“…The error on the first interval is bounded by (26) The error on the first interval is bounded by (26) …”

Section: Numerical Integrationmentioning

confidence: 99%

“…A quadrature approximation of the above integral is proposed in [16]. There are numerous other papers proposing exponentially convergent quadrature schemes for Dunford-Taylor integral representations of e −tL , e.g., [15,16,26]. Our third scheme below is an example of another exponentially convergent quadrature but, in our case, only involves multiple evaluations of (I + t i L) −1 f where t i ∈ (0, ∞) is related to the quadrature node.…”

Section: Introductionmentioning

confidence: 99%

“…Our third scheme below is an example of another exponentially convergent quadrature but, in our case, only involves multiple evaluations of (I + t i L) −1 f where t i ∈ (0, ∞) is related to the quadrature node. There are numerous other papers proposing exponentially convergent quadrature schemes for Dunford-Taylor integral representations of e −tL , e.g., [26,15,16].…”

Section: Introductionmentioning

confidence: 99%

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Numerical approximation of fractional powers of elliptic operators

2015

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We present and study a novel numerical algorithm to approximate the action of T β := L −β where L is a symmetric and positive definite unbounded operator on a Hilbert space H 0 . The numerical method is based on a representation formula for T −β in terms of Bochner integrals involving (I + t 2 L) −1 for t ∈ (0, ∞).To develop an approximation to T β , we introduce a finite element approximation L h to L and base our approximation to T β on T β h := L −β h . The direct evaluation of T β h is extremely expensive as it involves expansion in the basis of eigenfunctions for L h . The above mentioned representation formula holds for T −β h and we propose three quadrature approximations denoted generically by Q β h . The two results of this paper bound the errors in the H 0 inner product of T β − T β h π h and T β h − Q β h where π h is the H 0 orthogonal projection into the finite element space. We note that the evaluation of Q β h involves application of (I + (t i ) 2 L h ) −1 with t i being either a quadrature point or its inverse. Efficient solution algorithms for these problems are available and the problems at different quadrature points can be straightforwardly solved in parallel. Numerical experiments illustrating the theoretical estimates are provided for both the quadrature error T β h − Q β h and the finite element error T β − T β h π h .

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“…The error on the first interval is bounded by (26) The error on the first interval is bounded by (26) …”

Section: Numerical Integrationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical approximation of fractional powers of elliptic operators

2015

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“…Based on the techniques presented in [15,14,20,23,25], we next provide and study a sinc type quadrature method [22] for approximating u h avoiding the eigenfunction expansion in (1.5). We note that u h (t) = e −tL β h π h v with π h denoting the L 2 (Ω) projection onto H h .…”

Section: Introductionmentioning

confidence: 99%

The approximation of parabolic equations involving fractional powers of elliptic operators

Bonito

Lei

Pasciak

2017

Journal of Computational and Applied Mathematics

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We study the numerical approximation of a time dependent equation involving fractional powers of an elliptic operator L defined to be the unbounded operator associated with a Hermitian, coercive and bounded sesquilinear form on H 1 0 (Ω). The time dependent solution u(x, t) is represented as a Dunford Taylor integral along a contour in the complex plane.The contour integrals are approximated using sinc quadratures. In the case of homogeneous right-hand-sides and initial value v, the approximation results in a linear combination of functions (zqI − L) −1 v ∈ H 1 0 (Ω) for a finite number of quadrature points zq lying along the contour. In turn, these quantities are approximated using complex valued continuous piecewise linear finite elements.Our main result provides L 2 (Ω) error estimates between the solution u(·, t) and its final approximation. Numerical results illustrating the behavior of the algorithms are provided. arXiv:1607.07832v2 [math.NA]

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Iterative Solution of Shifted Positive-Definite Linear Systems Arising in a Numerical Method for the Heat Equation Based on Laplace Transformation and Quadrature

Cited by 2 publications

References 17 publications

Numerical approximation of fractional powers of elliptic operators

Numerical approximation of fractional powers of elliptic operators

The approximation of parabolic equations involving fractional powers of elliptic operators

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