2,907 research outputs found

    Numerical analysis of nonlinear subdiffusion equations

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    We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order α(0,1)\alpha\in(0,1) in time. The framework relies on three technical tools: a fractional version of the discrete Gr\"onwall-type inequality, discrete maximal regularity, and regularity theory of nonlinear equations. We establish a general criterion for showing the fractional discrete Gr\"onwall inequality, and verify it for the L1 scheme and convolution quadrature generated by BDFs. Further, we provide a complete solution theory, e.g., existence, uniqueness and regularity, for a time-fractional diffusion equation with a Lipschitz nonlinear source term. Together with the known results of discrete maximal regularity, we derive pointwise L2(Ω)L^2(\Omega) norm error estimates for semidiscrete Galerkin finite element solutions and fully discrete solutions, which are of order O(h2)O(h^2) (up to a logarithmic factor) and O(τα)O(\tau^\alpha), respectively, without any extra regularity assumption on the solution or compatibility condition on the problem data. The sharpness of the convergence rates is supported by the numerical experiments

    Pointwise best approximation results for Galerkin finite element solutions of parabolic problems

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    In this paper we establish a best approximation property of fully discrete Galerkin finite element solutions of second order parabolic problems on convex polygonal and polyhedral domains in the LL^\infty norm. The discretization method uses of continuous Lagrange finite elements in space and discontinuous Galerkin methods in time of an arbitrary order. The method of proof differs from the established fully discrete error estimate techniques and for the first time allows to obtain such results in three space dimensions. It uses elliptic results, discrete resolvent estimates in weighted norms, and the discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors in [16]. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish a local best approximation property that shows a more local behavior of the error at a given point

    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
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