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Adaptive Multirate Method for Coupled Parabolic and Elliptic Equations

Lee, Sara

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Abstract

In this thesis, motivated by the simulation of fuel cells and batteries, we develop an adaptive discretization algorithm to reduce the computational cost for solving the coupled parabolic/elliptic system. This system is the model for the electrochemical processes within the cathode of a solid oxide fuel cell (SOFC). First, the coupled system is discretized in time and in space by the Finite Element Method. Then, it is split into parabolic and elliptic sub-problems through an operator splitting method. These two equations are solved sequentially by the multirate iterative solving method that allows for different time step sizes for the temporal discretizations.

The main focus of this work is to derive goal-oriented, a posteriori error estimators based on the Dual Weighted Residual method that are computable and separately assess the temporal discretization error, the spatial discretization error and the splitting error for each sub-problem. Instead of natural norms, the errors are measured in an arbitrary quantity of interest, as is often used in practical applications.

The sub-problems are solved in temporal discretizations with different step lengths. If the ratio between the two step lengths is too large, this can result in the divergence of the coupling iteration within the multirate scheme. In this case, the algorithm uses the information from the splitting error estimator to control the convergence behavior. The error contributions of both discretizations and splitting method are balanced at the end of the refinement cycle that halts when the error estimators reach a desired accuracy.

The described methods are validated on a simplified model that simulates the cathode of a SOFC. In this application, the parabolic part consists of a reaction-diffusion equation describing the concentration distribution of ions, and the elliptic part describes electrical potential. For a given accuracy, the adaptive algorithm finds the least required number of degrees of freedom of the parabolic and the elliptic parts of the system. Since the electrical potential equation has the faster time scale, we use the multirate method and see that the elliptic problem requires a smaller number of degrees of freedom to attain the same desired accuracy within the system. This significantly saves the total computational cost, since the elliptic equation in the coupled system is more expensive to solve. Therefore, this combination of the degrees of freedoms is optimal, in that it gives the least computational cost and the convergence within the algorithm.

Item Type: Dissertation
Supervisor: Carraro, Dr. Thomas
Date of thesis defense: 3 November 2016
Date Deposited: 15 Nov 2016 14:29
Date: 2016
Faculties / Institutes: The Faculty of Mathematics and Computer Science > Department of Mathematics
The Faculty of Mathematics and Computer Science > Department of Applied Mathematics
Subjects: 500 Natural sciences and mathematics
510 Mathematics
Controlled Keywords: coupled system, multirate, parabolic equation, elliptic equation, splitting
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