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A variationally separable splitting for the generalized‐ α method for parabolic equations
Author(s) -
Behnoudfar Pouria,
Calo Victor M.,
Deng Quanling,
Minev Peter D.
Publication year - 2019
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.6246
Subject(s) - mathematics , discretization , separable space , partial differential equation , norm (philosophy) , solver , dissipation , tensor product , degrees of freedom (physics and chemistry) , parabolic partial differential equation , mathematical analysis , numerical analysis , mathematical optimization , physics , pure mathematics , quantum mechanics , political science , law , thermodynamics
We present a variationally separable splitting technique for the generalized‐ α method for solving parabolic partial differential equations. We develop a technique for a tensor‐product mesh which results in a solver with a linear cost with respect to the total number of degrees of freedom in the system for multidimensional problems. We consider finite elements and isogeometric analysis for the spatial discretization. The overall method maintains user‐controlled high‐frequency dissipation while minimizing unwanted low‐frequency dissipation. The method has second‐order accuracy in time and optimal rates ( h p +1 in L 2 norm of u and h p in L 2 norm of ∇ u ) in space. We present the spectral analysis on the amplification matrix to establish that the method is unconditionally stable. Various numerical examples illustrate the performance of the overall methodology and show the optimal approximation accuracy.