95 research outputs found

    Discrete maximal regularity of time-stepping schemes for fractional evolution equations

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    In this work, we establish the maximal p\ell^p-regularity for several time stepping schemes for a fractional evolution model, which involves a fractional derivative of order α(0,2)\alpha\in(0,2), α1\alpha\neq 1, in time. These schemes include convolution quadratures generated by backward Euler method and second-order backward difference formula, the L1 scheme, explicit Euler method and a fractional variant of the Crank-Nicolson method. The main tools for the analysis include operator-valued Fourier multiplier theorem due to Weis [48] and its discrete analogue due to Blunck [10]. These results generalize the corresponding results for parabolic problems

    A unified error analysis for spatial discretizations of wave-type equations with applications to dynamic boundary conditions

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    This thesis provides a unified framework for the error analysis of non-conforming space discretizations of linear wave equations in time-domain, which can be cast as symmetric hyperbolic systems or second-order wave equations. Such problems can be written as first-order evolution equations in Hilbert spaces with linear monotone operators. We employ semigroup theory for the well-posedness analysis and to obtain stability estimates for the space discretizations. To compare the finite dimensional approximations with the original solution, we use the concept of a lift from the discrete to the continuous space. Time integration with the Crank–Nicolson method is analyzed. In this framework, we derive a priori error bounds for the abstract space semi-discretization in terms of interpolation and discretization errors. These error bounds yield previously unkown convergence rates for isoparametric finite element discretizations of wave equations with dynamic boundary conditions in smooth domains. Moreover, our results allow to consider already investigated space discretizations in a unified way. Here it successfully reproduces known error bounds. Among the examples which we dicuss in this thesis are discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the scalar wave equation

    Additive domain decomposition operator splittings -- convergence analyses in a dissipative framework

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    We analyze temporal approximation schemes based on overlapping domain decompositions. As such schemes enable computations on parallel and distributed hardware, they are commonly used when integrating large-scale parabolic systems. Our analysis is conducted by first casting the domain decomposition procedure into a variational framework based on weighted Sobolev spaces. The time integration of a parabolic system can then be interpreted as an operator splitting scheme applied to an abstract evolution equation governed by a maximal dissipative vector field. By utilizing this abstract setting, we derive an optimal temporal error analysis for the two most common choices of domain decomposition based integrators. Namely, alternating direction implicit schemes and additive splitting schemes of first and second order. For the standard first-order additive splitting scheme we also extend the error analysis to semilinear evolution equations, which may only have mild solutions.Comment: Please refer to the published article for the final version which also contains numerical experiments. Version 3 and 4: Only comments added. Version 2, page 2: Clarified statement on stability issues for ADI schemes with more than two operator

    An Exponential Time Differencing Scheme with a Real Distinct Poles Rational Function for Advection-Diffusion Reaction Equations

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    A second order Exponential Time Differencing (ETD) scheme for advection-diffusion reaction systems is developed by using a real distinct poles rational function for approximating the underlying matrix exponential. The scheme is proved to be second order convergent. It is demonstrated to be robust for reaction-diffusion systems with non-smooth initial and boundary conditions, sharp solution gradients, and stiff reaction terms. In order to apply the scheme efficiently to higher dimensional problems, a dimensional splitting technique is also developed. This technique can be applied to all ETD schemes and has been found, in the test problems considered, to reduce computational time by up to 80%

    Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions

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    In this work, we propose novel discretisations of the spectral fractional Laplacian on bounded domains based on the integral formulation of the operator via the heat-semigroup formalism. Specifically, we combine suitable quadrature formulas of the integral with a finite element method for the approximation of the solution of the corresponding heat equation. We derive two families of discretisations with order of convergence depending on the regularity of the domain and the function on which the fractional Laplacian is acting. Unlike other existing approaches in literature, our method does not require the computation of the eigenpairs of the Laplacian on the considered domain, can be implemented on possibly irregular bounded domains, and can naturally handle different types of boundary constraints. Various numerical simulations are provided to illustrate performance of the proposed method and support our theoretical results.FdT acknowledges support of Toppforsk project Waves and Nonlinear Phenomena (WaNP), grant no. 250070 from the Research Council of Norway. ERCIM ``Alain Benoussan" Fellowship programm
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