
A linear implicit Euler method for the finite element discretization of a controlled stochastic heat equation
We consider a numerical approximation of a linear quadratic control prob...
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A general framework of low regularity integrators
We introduce a new general framework for the approximation of evolution ...
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A Numerical Method for a Nonlocal Diffusion Equation with Additive Noise
We consider a nonlocal evolution equation representing the continuum lim...
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On the wellposedness of the Galerkin semidiscretization of the periodic initialvalue problem of the Serre equations
We consider the periodic initialvalue problem for the Serre equations o...
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Exponential integrators for semilinear parabolic problems with linear constraints
This paper is devoted to the construction of exponential integrators of ...
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Weighted error estimates for transient transport problems discretized using continuous finite elements with interior penalty stabilization on the gradient jumps
In this paper we consider the semidiscretization in space of a first or...
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Time discretization of an initial value problem for a simultaneous abstract evolution equation applying to parabolichyperbolic phasefield systems
This article deals with a simultaneous abstract evolution equation. This...
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Exponential Runge Kutta time semidiscetizations with low regularity initial data
We apply exponential Runge Kutta time discretizations to semilinear evolution equations d U/ d t=AU+B(U) posed on a Hilbert space Y. Here A is normal and generates a strongly continuous semigroup, and B is assumed to be a smooth nonlinearity from Y_ℓ = D(A^ℓ) to itself, and ℓ∈ I ⊆ [0,L], L ≥ 0, 0,L ∈ I. In particular the semilinear wave equation and nonlinear Schrödinger equation with periodic boundary conditions or posed on R^d fit into this framework. We prove convergence of order O(h^min(ℓ,p)) for nonsmooth initial data U^0∈ Y_ℓ, where ℓ >0, for a method of classical order p. We show in an example of an exponential Euler discretization of a linear evolution equation that our estimates are sharp, and corroborate this in numerical experiments for a semilinear wave equation. To prove our result we Galerkin truncate the semiflow and numerical method and balance the Galerkin truncation error with the error of the time discretization of the projected system. We also extend these results to exponential Rosenbrock methods.
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