A spectral element method for solving parabolic initial boundary value problems on smooth domains using parallel computers is presented in this paper. The space domain is divided into a number of shape regular quadrilaterals of size h and the time step k is proportional to h2. At each time step we minimize a functional which is the sum of the squares of the residuals in the partial differential equation, initial condition and boundary condition in different Sobolev norms and a term which measures the jump in the function and its derivatives across inter-element boundaries in certain Sobolev norms. The Sobolev spaces used are of different orders in space and time. We can define a preconditioner for the minimization problem which allows the problem to decouple. Error estimates are obtained for both the h and p versions of this method. © 2006 Elsevier B.V. All rights reserved.
CITATION STYLE
Kopriva, D. A. (2009). Spectral Element Methods (pp. 293–354). https://doi.org/10.1007/978-90-481-2261-5_8
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