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Multigrid method

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2017, Master of Science, Ohio State University, Mathematics.
The multigrid method is an algorithm aiming to provide a numerical solution to a system of equations. In this thesis we focus on its original purpose: solving differential equations. By discretizing a differential equation, one obtains a system of linear equations, which can be solved using multigrid methods. We will start our discussion in one dimension, considering ordinary differential equations. First, we will review standard pointwise relaxation methods, since they are a fundamental part of any multigrid method. We will compare them in different settings and see their shortcomings. This part also provides an insight of the need for a better method, and one of the better methods is the multigrid method. Then we will discuss elements of multigrid method in one dimension. We will apply the method on different problems, compare and discuss the results, providing proofs and calculations for easy but well-known phenomena, such as the reduction rate of the error and the exact outcome of the method on the homogeneous Poisson equation when damped Jacobi relaxation is chosen to be used in the multigrid method, and proof of the fact that the error for the same equation can be totally eliminated in one single iteration using the multigrid method with the choice of red-black Gauss-Seidel relaxation. We will consider a nontrivial ordinary differential equation in order to be convinced that the problems that the multigrid method can deal with and the conclusions we have arrived can be generalized. Afterwards we discuss the multigrid method in two dimensions, where it is mostly used in real life applications. As we did in one dimension, we shortly review the relaxation methods in two dimensions. Then we introduce the elements of the multigrid method in two dimensions. We apply the multigrid method to some problems and tabulate the results, leaving the redundant cases out. We discuss the results and compare them with the one dimensional results.
Edward Overman (Advisor)
79 p.

Recommended Citations

Citations

  • Senel, G. (2017). Multigrid method [Master's thesis, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1494347278971509

    APA Style (7th edition)

  • Senel, Gunes. Multigrid method. 2017. Ohio State University, Master's thesis. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1494347278971509.

    MLA Style (8th edition)

  • Senel, Gunes. "Multigrid method." Master's thesis, Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1494347278971509

    Chicago Manual of Style (17th edition)