This paper is concerned with the accurate computational error estimation of numerical solutions of multi-scale, multi-physics systems of reaction-diffusion equations. Such systems can present significantly different t...
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This paper is concerned with the accurate computational error estimation of numerical solutions of multi-scale, multi-physics systems of reaction-diffusion equations. Such systems can present significantly different temporal and spatial scales within the components of the model, indicating the use of independent discretizations for different components. However, multi-discretization can have significant effects on accuracy and stability. We perform an adjoint-based analysis to derive asymptotically accurate a posteriori error estimates for a user-defined quantity of interest. These estimates account for leading order contributions to the error arising from numerical solution of each component, an error due to incomplete iteration, an error due to linearization, and for errors arising due to the projection of solution components between different spatial meshes. Several numerical examples with various settings are given to demonstrate the performance of the error estimators. (C) 2013 Elsevier B.V. All rights reserved.
This paper concerns the development of a novel multi-level computational tool for simulations of very large-scale problems arising in science and technology. One of the particular applications can be numerical simulat...
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This paper concerns the development of a novel multi-level computational tool for simulations of very large-scale problems arising in science and technology. One of the particular applications can be numerical simulations of material properties such as effective thermal diffusivity and/or effective Young's moduli of nanocomposites reinforced by carbon nanotubes. Here we present the multi-level boundary element method (MLBEM) for solutions of very large thermal problems, and focus on efficient solutions of steady heat diffusion. First, we perform analyses of numerical error and computational complexity for the multi-level boundary element algorithm and show that the optimal complexity of the algorithm is O(N log N). Next, we consider a model problem of line multi-integral evaluation and investigate the performance of the MLBEM formulation using a single-patch approach. Then, we study the performance of the multi-level boundary element formulation on an example Neumann problem of steady heat diffusion leading to a boundary integral equation of the second kind. Here, we solve a problem involving four million degrees of freedom in less than one hour on a desk-top workstation. Next, we consider a model problem in a unit square with mixed boundary conditions and study the performance for the new MLBEM formulation. Finally, we consider an example problem of heat conduction in composite material with the heat conductivity ratio of 100:1 for fiber elements and a matrix, and study effective conductivity for volume fraction up to 3%.
Frictional dynamic contact problems with complex geometries are a challenging task - from the computational as well as from the analytical point of view - since they generally involve space and time multi-scale aspect...
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Frictional dynamic contact problems with complex geometries are a challenging task - from the computational as well as from the analytical point of view - since they generally involve space and time multi-scale aspects. To be able to reduce the complexity of this kind of contact problem, we employ a non-conforming domain decomposition method in space, consisting of a coarse global mesh not resolving the local structure and an overlapping fine patch for the contact computation. This leads to several benefits: First, we resolve the details of the surface only where it is needed, i.e., in the vicinity of the actual contact zone. Second, the subproblems can be discretized independently of each other which enables us to choose a much finer time scale on the contact zone than on the coarse domain. Here, we propose a set of interface conditions that yield optimal a priori error estimates on the fine-meshed subdomain without any artificial dissipation. Further, we develop an efficient iterative solution scheme for the coupled problem that is robust with respect to jumps in the material parameters. Several complex numerical examples illustrate the performance of the new scheme. (C) 2011 Elsevier B.V. All rights reserved.
This paper proposes and analyzes an a posteriori error estimator for the finite element multi-scale discretization approximation of the Steklov eigenvalue problem. Based on the a posteriori error estimates, an adaptiv...
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This paper proposes and analyzes an a posteriori error estimator for the finite element multi-scale discretization approximation of the Steklov eigenvalue problem. Based on the a posteriori error estimates, an adaptive algorithm of shifted inverse iteration type is designed. Finally, numerical experiments comparing the performances of three kinds of different adaptive algorithms are provided, which illustrate the efficiency of the adaptive algorithm proposed here. (C) 2016 IMACS. Published by Elsevier B.V. All rights reserved.
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