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    On periodic solutions for one-phase and two-phase problems of the Navier–Stokes equations
    (Basel : Springer, 2020) Eiter, Thomas; Kyed, Mads; Shibata, Yoshihiro
    This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier–Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasilinear systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 0, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal Lp–Lq regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, which is obtained by the systematic use of R-solvers developed in Shibata (Diff Int Eqns 27(3–4):313–368, 2014; On the R-bounded solution operators in the study of free boundary problem for the Navier–Stokes equations. In: Shibata Y, Suzuki Y (eds) Springer proceedings in mathematics & statistics, vol. 183, Mathematical Fluid Dynamics, Present and Future, Tokyo, Japan, November 2014, pp 203–285, 2016; Comm Pure Appl Anal 17(4): 1681–1721. https://doi.org/10.3934/cpaa.2018081, 2018; R boundedness, maximal regularity and free boundary problems for the Navier Stokes equations, Preprint 1905.12900v1 [math.AP] 30 May 2019) to the resolvent problem for the linearized equations and the transference theorem obtained in Eiter et al. (R-solvers and their application to periodic Lp estimates, Preprint in 2019) for the Lp boundedness of operator-valued Fourier multipliers. These approaches are the novelty of this paper. © 2020, The Author(s).
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    Approximation of solutions to multidimensional parabolic equations by approximate approximations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Lanzara, Flavia; Mazya, Vladimir; Schmidt, Gunther
    We propose a fast method for high order approximations of the solution of n-dimensional parabolic problems over hyper-rectangular domains in the framework of the method of approximate approximations. This approach, combined with separated representations, makes our method effective also in very high dimensions.We report on numerical results illustrating that our formulas are accurate and provide the predicted approximation rate 6 also in high dimensions.
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    Stability of explicit Runge-Kutta methods for high order finite element approximation of linear parabolic equations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Huang, Weizhang; Kamenski, Lennard; Lang, Jens
    We study the stability of explicit Runge-Kutta methods for high order Lagrangian finite element approximation of linear parabolic equations and establish bounds on the largest eigenvalue of the system matrix which determines the largest permissible time step. A bound expressed in terms of the ratio of the diagonal entries of the stiffness and mass matrices is shown to be tight within a small factor which depends only on the dimension and the choice of the reference element and basis functions but is independent of the mesh or the coefficients of the initial-boundary value problem under consideration. Another bound, which is less tight and expressed in terms of mesh geometry, depends only on the number of mesh elements and the alignment of the mesh with the diffusion matrix. The results provide an insight into how the interplay between the mesh geometry and the diffusion matrix affects the stability of explicit integration schemes when applied to a high order finite element approximation of linear parabolic equations on general nonuniform meshes.
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    Stability of explicit Runge-Kutta methods for finite element approximation of linear parabolic equations on anisotropic meshes
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Huang, Weizhang; Kamenski, Lennard; Lang, Jens
    We study the stability of explicit Runge-Kutta integration schemes for the linear finite element approximation of linear parabolic equations. The derived bound on the largest permissible time step is tight for any mesh and any diffusion matrix within a factor of 2(d + 1), where d is the spatial dimension. Both full mass matrix and mass lumping are considered. The bound reveals that the stability condition is affected by two factors. The first one depends on the number of mesh elements and corresponds to the classic bound for the Laplace operator on a uniform mesh. The other factor reflects the effects of the interplay of the mesh geometry and the diffusion matrix. It is shown that it is not the mesh geometry itself but the mesh geometry in relation to the diffusion matrix that is crucial to the stability of explicit methods. When the mesh is uniform in the metric specified by the inverse of the diffusion matrix, the stability condition is comparable to the situation with the Laplace operator on a uniform mesh. Numerical results are presented to verify the theoretical findings.
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    Exact controllability on a curve for a semilinear parabolic equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Hömberg, Dietmar; Yamamoto, Masahiro
    Motivated by the growing number of industrially important laser material treatments we investigate the controllability on a curve for a semilinear parabolic equation. We prove the local exact controllability and a global stability result in the twodimensional setting. As an application we consider the control of laser surface hardening. We show that our theory applies to this situation and present numerical simulations for a PID control of laser hardening. Moreover, the result of an industrial case study is presented confirming that the numerically derived temperature in the hot-spot of the laser can indeed be used as set-point for the machine-based process control.