We consider Galerkin approximation in space of linear parabolic initialboundary value problems where the elliptic operator is symmetric and thus induces an energy norm. For two related variational settings, we show th...
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ISBN:
(纸本)9783319399294;9783319399270
We consider Galerkin approximation in space of linear parabolic initialboundary value problems where the elliptic operator is symmetric and thus induces an energy norm. For two related variational settings, we show that the quasioptimality constant equals the stability constant of the L-2- projection with respect to that energy norm.
In this paper we review various numerical homogenization methods for monotone parabolic problems with multiple scales. The spatial discretisation is based on finite element methods and the multiscale strategy relies o...
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ISBN:
(纸本)9783319416403;9783319416380
In this paper we review various numerical homogenization methods for monotone parabolic problems with multiple scales. The spatial discretisation is based on finite element methods and the multiscale strategy relies on the heterogeneous multiscale method. The time discretization is performed by several classes of Runge-Kutta methods (strongly A-stable or explicit stabilized methods). We discuss the construction and the analysis of such methods for a range of problems, from linear parabolic problems to nonlinear monotone parabolic problems in the very general L-p(W-1,W-p) setting. We also show that under appropriate assumptions, a computationally attractive linearized method can be constructed for nonlinear problems.
In this paper, we show a local a priori error estimate for the Poisson equation in three space dimensions (3D), where the source term is a Dirac measure concentrated on a line. This type of problem can be found in man...
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ISBN:
(纸本)9783319399294;9783319399270
In this paper, we show a local a priori error estimate for the Poisson equation in three space dimensions (3D), where the source term is a Dirac measure concentrated on a line. This type of problem can be found in many application areas. In medical engineering, e.g., blood flow in capillaries and tissue can be modeled by coupling Poiseuille's and Darcy's law using a line source term. Due to the singularity induced by the line source term, finite element solutions converge suboptimal in classical norms. However, quite often the error at the singularity is either dominated by model errors (e.g. in dimension reduced settings) or is not the quantity of interest (e.g. in optimal control problems). Therefore we are interested in local error estimates, i.e., we consider in space a L-2-norm on a fixed subdomain excluding a neighborhood of the line, where the Dirac measure is concentrated. It is shown that linear finite elements converge optimal up to a log-factor in such a norm. The theoretical considerations are confirmed by some numerical tests.
A comparison study of different decoupled schemes for the evolutionary Stokes/Darcy problem is carried out. Stability and error estimates of a mass conservative multiple-time-step algorithm are provided under a time s...
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In this study, possibility of reducing drag in turbulent pipe flow via phase randomization is investigated. Phase randomization is a passive drag reduction mechanism, the main idea behind which is, reduction in drag c...
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ISBN:
(纸本)9783319399294;9783319399270
In this study, possibility of reducing drag in turbulent pipe flow via phase randomization is investigated. Phase randomization is a passive drag reduction mechanism, the main idea behind which is, reduction in drag can be obtained via distrupting the wave-like structures present in the flow. To facilitate the investigation flow in a circular cylindrical pipe is simulated numerically. DNS (direct numerical simulation) approach is used with a solenoidal spectral formulation, hence the continuity equation is automatically satisfied (Tugluk and Tarman, Acta Mech 223(5): 921-935, 2012). Simulations are performed for flow driven by a constant mass flux, at a bulk Reynolds number (Re) of 4900. Legendre polynomials are used in constructing the solenoidal basis functions employed in the numerical method.
We present a novel data-driven approach to propagate uncertainty. It consists of a highly efficient integrated adaptive sparse grid approach. We remove the gap between the subjective assumptions of the input's unc...
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ISBN:
(数字)9783319282626
ISBN:
(纸本)9783319282626;9783319282602
We present a novel data-driven approach to propagate uncertainty. It consists of a highly efficient integrated adaptive sparse grid approach. We remove the gap between the subjective assumptions of the input's uncertainty and the unknown real distribution by applying sparse grid density estimation on given measurements. We link the estimation to the adaptive sparse grid collocation method for the propagation of uncertainty. This integrated approach gives us two main advantages: First, the linkage of the density estimation and the stochastic collocation method is straightforward as they use the same fundamental principles. Second, we can efficiently estimate moments for the quantity of interest without any additional approximation errors. This includes the challenging task of solving higher-dimensional integrals. We applied this newapproach to a complex subsurface flow problem and showed that it can compete with state-of-the-art methods. Our sparse grid approach excels by efficiency, accuracy and flexibility and thus can be applied in many fields from financial to environmental sciences.
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