We present a numerical method for the solution of integro-differential equations describing motion of an incompressible viscoelastic fluid with memory. In particular, the system of equations consists of the momentum c...
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ISBN:
(纸本)9783319399294;9783319399270
We present a numerical method for the solution of integro-differential equations describing motion of an incompressible viscoelastic fluid with memory. In particular, the system of equations consists of the momentum conservation equation with the Cauchy stress tensor divided in a viscous and an elastic parts which depend non-linearly on the symmetric part of velocity gradient and nonlinearly on the past values of the Finger strain tensor, respectively. The momentum conservation equation is completed with system of equations that describes relation between the velocity gradient and the Finger strain tensor. The method is based on a discontinuous Galerkin method in the spatial variables and the BDF methods in the time variables.
In this contribution we are concerned with efficient model reduction for multiscale problems arising in lithium-ion battery modeling with spatially resolved porous electrodes. We present new results on the application...
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ISBN:
(纸本)9783319399294;9783319399270
In this contribution we are concerned with efficient model reduction for multiscale problems arising in lithium-ion battery modeling with spatially resolved porous electrodes. We present new results on the application of the reduced basis method to the resulting instationary 3D battery model that involves strong non-linearities due to Buttler-Volmer kinetics. Empirical operator interpolation is used to efficiently deal with this issue. Furthermore, we present the localized reduced basis multiscale method for parabolic problems applied to a thermal model of batteries with resolved porous electrodes. Numerical experiments are given that demonstrate the reduction capabilities of the presented approaches for these real world applications.
A (multi-) wavelet expansion is used to derive a rigorous bound for the (dual) norm Reduced Basis residual. We show theoretically and numerically that the error estimator is online efficient, reliable and rigorous. It...
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ISBN:
(纸本)9783319399294;9783319399270
A (multi-) wavelet expansion is used to derive a rigorous bound for the (dual) norm Reduced Basis residual. We show theoretically and numerically that the error estimator is online efficient, reliable and rigorous. It allows to control the exact error (not only with respect to a "truth" discretization).
In this paper, we review and refine the main ideas for devising the so-called hybridizable discontinuous Galerkin (HDG) methods;we do that in the framework of steady-state diffusion problems. We begin by revisiting th...
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ISBN:
(纸本)9783319416403;9783319416380
In this paper, we review and refine the main ideas for devising the so-called hybridizable discontinuous Galerkin (HDG) methods;we do that in the framework of steady-state diffusion problems. We begin by revisiting the classic techniques of static condensation of continuous finite element methods and that of hybridization of mixed methods, and show that they can be reinterpreted as discrete versions of a characterization of the associated exact solution in terms of solutions of Dirichlet boundary-value problems on each element of the mesh which are then patched together by transmission conditions across interelement boundaries. We then define the HDG methods associated to this characterization as those using discontinuous Galerkin (DG) methods to approximate the local Dirichlet boundary-value problems, and using weak impositions of the transmission conditions. We give simple conditions guaranteeing the existence and uniqueness of their approximate solutions, and show that, by their very construction, the HDG methods are amenable to static condensation. We do this assuming that the diffusivity tensor can be inverted;we also briefly discuss the case in which it cannot. We then show how a different characterization of the exact solution, gives rise to a different way of statically condensing an already known HDG method. We devote the rest of the paper to establishing bridges between the HDG methods and other methods (the old DG methods, the mixed methods, the staggered DG method and the so-called Weak Galerkin method) and to describing recent efforts for the construction of HDG methods (one for systematically obtaining superconvergent methods and another, quite different, which gives rise to optimally convergent methods). We end by providing a few bibliographical notes and by briefly describing ongoing work.
European basket options are priced by solving the multi-dimensional Black-Scholes-Merton equation. Standard numerical methods to solve these problems often suffer from the "curse of dimensionality". We tackl...
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ISBN:
(纸本)9783319399294;9783319399270
European basket options are priced by solving the multi-dimensional Black-Scholes-Merton equation. Standard numerical methods to solve these problems often suffer from the "curse of dimensionality". We tackle this by using a dimension reduction technique based on a principal component analysis with an asymptotic expansion. Adaptive finite differences are used for the spatial discretization. In time we employ a discontinuous Galerkin scheme. The efficiency of our proposed method to solve a five-dimensional problem is demonstrated through numerical experiments and compared with a Monte-Carlo method.
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