A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature
@article{Sheen2003APM, title={A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature}, author={Dongwoo Sheen and Ian H. Sloan and Vidar Thom{\'e}e}, journal={Ima Journal of Numerical Analysis}, year={2003}, volume={23}, pages={269-299} }
We consider the discretization in time of an inhomogeneous parabolic equation in a Banach space setting, using a representation of the solution as an integral along a smooth curve in the complex left half-plane which, after transformation to a finite interval, is then evaluated to high accuracy by a quadrature rule. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. The paper is a further development of earlier work by the…
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References
SHOWING 1-10 OF 17 REFERENCES
A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature
- MathematicsMath. Comput.
- 2000
We treat the time discretization of an initial-value problem for a homogeneous abstract parabolic equation by first using a representation of the solution as an integral along the boundary of a…
Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term
- MathematicsMath. Comput.
- 1996
The proposed discretization uses convolution quadrature based on the first- and second-order backward difference methods in time, and piecewise linear finite elements in space to study the numerical approximation of an integro-differential equation.
Maximum-norm estimates for resolvents of elliptic finite element operators
- MathematicsMath. Comput.
- 2003
It is shown directly that such a resolvent estimate of standard form in the maximum-norm outside some sector in the right halfplane holds outside any sector around the positive real axis, with arbitrarily small angle, useful in the study of fully discrete approximations based on A(θ)-stable rational functions.
Generation of analytic semigroups by strongly elliptic operators
- Mathematics
- 1974
Strongly elliptic operators are shown to generate analytic semigroups of evolution operators in the topology of uniform convergence, when realized under general boundary conditions on (possibly)…
Stability and analyticity in maximum-norm for simplicial Lagrange finite element semidiscretizations of parabolic equations with Dirichlet boundary conditions
- MathematicsNumerische Mathematik
- 2000
Abstract. Stability and analyticity estimates in maximum-norm are shown for spatially discrete finite element approximations based on simplicial Lagrange elements for the model heat equation with…
The Accurate Numerical Inversion of Laplace Transforms
- Mathematics
- 1979
Inversion of almost arbitrary Laplace transforms is effected by trapezoidal integration along a special contour. The number n of points to be used is one of several parameters, in most cases yielding…
The Laplace Transform
- MathematicsNature
- 1943
THE theory of Fourier integrals arises out of the elegant pair of reciprocal formulæThe Laplace TransformBy David Vernon Widder. (Princeton Mathematical Series.) Pp. x + 406. (Princeton: Princeton…
Introduction to Numerical Analysis
- Mathematics
- 2001
1. The numerical evaluation of expressions 2. Linear systems of equations 3. Interpolation and numerical differentiation 4. Numerical integration 5. Univariate nonlinear equations 6. Systems of…