In this work we present an iterative coupling scheme for the quasi-static Biot system of poroelasticity. For the discretization of the subproblems describing mechanical deformation and single-phase flow space-time fin...
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ISBN:
(纸本)9783319399294;9783319399270
In this work we present an iterative coupling scheme for the quasi-static Biot system of poroelasticity. For the discretization of the subproblems describing mechanical deformation and single-phase flow space-time finite element methods based on a discontinuous Galerkin approximation of the time variable are used. The spatial approximation of the flow problem is done by mixed finite element methods. The stability of the approach is illustrated by numerical experiments. The presented variational space-time framework is of higher order accuracy such that problems with high fluctuations become feasible. Moreover, it offers promising potential for the simulation of the fully dynamic Biot-Allard system coupling an elastic wave equation for solid's deformation with single-phase flow for fluid infiltration.
Projection based variational multiscale (VMS) methods are a very successful technique in the numerical simulation of high Reynolds number flow problems using coarse discretizations. However, their implementation into ...
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ISBN:
(纸本)9783319399294;9783319399270
Projection based variational multiscale (VMS) methods are a very successful technique in the numerical simulation of high Reynolds number flow problems using coarse discretizations. However, their implementation into an existing (legacy) codes can be very challenging in practice. We propose a second order variant of projection-based VMS method for non-isothermal flow problems. The method adds stabilization as a decoupled post-processing step for both velocity and temperature, and thus can be efficiently and easily used with existing codes. In this work, we propose the algorithm and give numerical results for convergence rates tests and coarse mesh simulation of Marsigli flow.
In this paper we discuss the adjoint stabilised finite element method introduced in Burman (SIAM J Sci Comput 35( 6): A2752-A2780, 2013) and how it may be used for the computation of solutions to problems for which th...
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ISBN:
(纸本)9783319416403;9783319416380
In this paper we discuss the adjoint stabilised finite element method introduced in Burman (SIAM J Sci Comput 35( 6): A2752-A2780, 2013) and how it may be used for the computation of solutions to problems for which the standard stability theory given by the Lax-Milgram Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to ill-posed problems that have some conditional stability property and prove (conditional) error estimates in an abstract framework. As a model problem we consider the elliptic Cauchy problem and provide a complete numerical analysis for this case. Some numerical examples are given to illustrate the theory.
The sparse grid combination technique provides a framework to solve high-dimensional numerical problems with standard solvers by assembling a sparse grid from many coarse and anisotropic full grids called component gr...
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ISBN:
(数字)9783319282626
ISBN:
(纸本)9783319282626;9783319282602
The sparse grid combination technique provides a framework to solve high-dimensional numerical problems with standard solvers by assembling a sparse grid from many coarse and anisotropic full grids called component grids. Hierarchization is one of the most fundamental tasks for sparse grids. It describes the transformation from the nodal basis to the hierarchical basis. In settings where the component grids have to be frequently combined and distributed in a massively parallel compute environment, hierarchization on component grids is relevant to minimize communication overhead. We present a cache-oblivious hierarchization algorithm for component grids of the combination technique. It causes vertical bar G(l)vertical bar . (1/B + O (1/(d)root M)) cache misses under the tall cache assumption M = omega (B-d).(1) Here, G(l) denotes the component grid, d the dimension, M the size of the cache and B the cache line size. This algorithm decreases the leading term of the cache misses by a factor of d compared to the unidirectional algorithm which is the common standard up to now. The new algorithm is also optimal in the sense that the leading term of the cache misses is reduced to scanning complexity, i.e., every degree of freedom has to be touched once. We also present a variant of the algorithm that causes vertical bar G(l)vertical bar . (2/B + O (1/(d-1)root ***-2)) cache misses under the assumption M = omega (B). The new algorithms have been implemented and outperform previously existing software. In several cases the measured performance is close to the best possible.
The Vlasov-Poisson equation models the evolution of a plasma in an external or self-consistent electric field. The model consists of an advection equation in six dimensional phase space coupled to Poisson's equati...
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ISBN:
(数字)9783319282626
ISBN:
(纸本)9783319282626;9783319282602
The Vlasov-Poisson equation models the evolution of a plasma in an external or self-consistent electric field. The model consists of an advection equation in six dimensional phase space coupled to Poisson's equation. Due to the high dimensionality and the development of small structures the numerical solution is quite challenging. For two or four dimensional Vlasov problems, semi-Lagrangian solvers have been successfully applied. Introducing a sparse grid, the number of grid points can be reduced in higher dimensions. In this paper, we present a semi-Lagrangian Vlasov-Poisson solver on a tensor product of two sparse grids. In order to defeat the problem of poor representation of Gaussians on the sparse grid, we introduce a multiplicative delta-f method and separate a Gaussian part that is then handled analytically. In the semi-Lagrangian setting, we have to evaluate the hierarchical surplus on each mesh point. This interpolation step is quite expensive on a sparse grid due to the global nature of the basis functions. In our method, we use an operator splitting so that the advection steps boil down to a number of one dimensional interpolation problems. With this structure in mind we devise an evaluation algorithm with constant instead of logarithmic complexity per grid point. Results are shown for standard test cases and in four dimensional phase space the results are compared to a full-grid solution and a solution on the four dimensional sparse grid.
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