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Item type:Publication, Access status: with files , Pathwise uniform convergence of time discretization schemes for SPDEsIn this paper we prove convergence rates for time discretization schemes for semilinear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator is the generator of a strongly continuous semigroup on a Hilbert space, and the focus is on nonparabolic problems. The main results are optimal bounds for the uniform strong errorwhere, is the mild solution, is obtained from a time discretization scheme, is the step size and. The usual schemes such as the exponential Euler (EE), the implicit Euler (IE), the Crank-Nicolson (CN) method, etc. are included as special cases. Under conditions on the nonlinearity and the noise, we show (linear equation, additive noise, general) (nonlinear equation, multiplicative noise, contractive) (nonlinear wave equation, multiplicative noise), for a large class of time discretization schemes. The logarithmic factor can be removed if the EE method is used with a (quasi)-contractive. The obtained bounds coincide with the optimal bounds for SDEs. Most of the existing literature is concerned with bounds for the simpler pointwise strong errorApplications to Maxwell equations, Schrödinger equations and wave equations are included. For these equations, our results improve and reprove several existing results with a unified method and provide the first results known for the IE and the CN method.Publicationtype: Journal Article12 35 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Auxiliary space preconditioners for SIP-DG discretizations of H(curl)-elliptic problems with discontinuous coefficientsWe propose a family of preconditioners for linear systems of equations arising from a piecewise polynomial symmetric interior penalty discontinuous Galerkin discretization of H(curl,ω)-elliptic boundary value problems on conforming meshes. The design and analysis of the proposed preconditioners rely on the auxiliary space method (ASM) employing an auxiliary space of H(curl,ω)-conforming finite element functions together with a relaxation technique (local smoothing). On simplicial meshes, the proposed preconditioner enjoys asymptotic optimality with respect to mesh refinement. It is also robust with respect to jumps in the coefficients ? and b in the second-and zeroth-order parts of the operator, respectively, except when the problem changes from curl-dominated to reaction-dominated and vice versa. On quadrilateral/hexahedral meshes some of the proposed ASM solvers may fail, since the related H(curl,ω)-conforming finite element space does not provide a spectrally accurate discretization. Extensive numerical experiments are included to verify the theory and assess the performance of the preconditioners.Publicationtype: Journal Article1072 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Optimal scaling for p-norms and componentwise distance to singularityWe give lower and upper bounds for the optimal p-norm condition number achievable by two-sided diagonal scalings. There are no assumptions on the irreducibility of certain matrices. The bounds are shown to be optimal for the 2-norm. For the 1-norm and inf-norm the (known) exact value of the optimal condition number is confirmed. We give means how to calculate the minimizing diagonal matrices. Furthermore, a class of new lower bounds for the componentwise distance to the nearest singular matrix is given. They are shown to be superior to existing ones.Publicationtype: Journal Article32