Stohrer, Christian. Finite element heterogeneous multiscale methods for the wave equation. 2013, Doctoral Thesis, University of Basel, Faculty of Science.

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Official URL: http://edoc.unibas.ch/diss/DissB_10425
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Abstract
Wave phenomena appear in a wide range of applications such as fullwaveform seismic inversion, medical imaging, or composite materials. Often, they are modeled by the acoustic wave equation.
It can be solved by standard numerical methods such as, e.g., the finite element (FE) or the finite difference method. However, if the wave propagation speed varies on a microscopic length scale denoted by epsilon, the computational cost becomes infeasible, since the medium must be resolved down to its finest scale. In this thesis we propose multiscale numerical methods which approximate the overall macroscopic behavior of the wave propagation with a substantially lower computational effort. We follow the design principles of the heterogeneous multiscale method (HMM), introduced in 2003 by E and Engquist. This method relies on a coarse discretization of an a priori unknown effective equation. The missing data, usually the parameters of the effective equation, are estimated on demand by solving microscale problems on small sampling domains. Hence, no precomputation of these effective parameters is needed. We choose FE methods to solve both the macroscopic and the microscopic problems.
For limited time the overall behavior of the wave is well described by the homogenized wave equation. We prove that the FEHMM method converges to the solution of the homogenized wave equation. With increasing time, however, the true solution deviates from the classical homogenization limit, as a large secondary wave train develops. Neither the homogenized solution, nor the FEHMM capture these dispersive effects. To capture them we need to modify the FEHMM. Inspired by higher order homogenization techniques we additionally compute a correction term of order epsilon^2. Since its computation also relies on the solution of the same microscale problems as the original FEHMM, the computational effort remains essentially unchanged. For this modified version we also prove convergence to the homogenized wave equation, but in contrast to the original FEHMM the longtime dispersive behavior is recovered.
The convergence proofs for the FEHMM follow from new Strangtype results for the wave equation. The results are general enough such that the FEHMM with and without the longtime correction fits into the setting, even if numerical quadrature is used to evaluate the arising L^2 inner product.
In addition to these results we give alternative formulations of the FEHMM, where the elliptic micro problems are replaced by hyperbolic ones. All the results are supported by numerical tests. The versatility of the method is demonstrated by various numerical examples.
It can be solved by standard numerical methods such as, e.g., the finite element (FE) or the finite difference method. However, if the wave propagation speed varies on a microscopic length scale denoted by epsilon, the computational cost becomes infeasible, since the medium must be resolved down to its finest scale. In this thesis we propose multiscale numerical methods which approximate the overall macroscopic behavior of the wave propagation with a substantially lower computational effort. We follow the design principles of the heterogeneous multiscale method (HMM), introduced in 2003 by E and Engquist. This method relies on a coarse discretization of an a priori unknown effective equation. The missing data, usually the parameters of the effective equation, are estimated on demand by solving microscale problems on small sampling domains. Hence, no precomputation of these effective parameters is needed. We choose FE methods to solve both the macroscopic and the microscopic problems.
For limited time the overall behavior of the wave is well described by the homogenized wave equation. We prove that the FEHMM method converges to the solution of the homogenized wave equation. With increasing time, however, the true solution deviates from the classical homogenization limit, as a large secondary wave train develops. Neither the homogenized solution, nor the FEHMM capture these dispersive effects. To capture them we need to modify the FEHMM. Inspired by higher order homogenization techniques we additionally compute a correction term of order epsilon^2. Since its computation also relies on the solution of the same microscale problems as the original FEHMM, the computational effort remains essentially unchanged. For this modified version we also prove convergence to the homogenized wave equation, but in contrast to the original FEHMM the longtime dispersive behavior is recovered.
The convergence proofs for the FEHMM follow from new Strangtype results for the wave equation. The results are general enough such that the FEHMM with and without the longtime correction fits into the setting, even if numerical quadrature is used to evaluate the arising L^2 inner product.
In addition to these results we give alternative formulations of the FEHMM, where the elliptic micro problems are replaced by hyperbolic ones. All the results are supported by numerical tests. The versatility of the method is demonstrated by various numerical examples.
Advisors:  Grote, Marcus J. 

Committee Members:  Runborg, Olof 
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Numerik (Grote) 
UniBasel Contributors:  Grote, Marcus J. 
Item Type:  Thesis 
Thesis Subtype:  Doctoral Thesis 
Thesis no:  10425 
Thesis status:  Complete 
Number of Pages:  98 S. 
Language:  English 
Identification Number: 

Last Modified:  22 Jan 2018 15:51 
Deposited On:  18 Jul 2013 10:11 
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