scholarly journals A Mathematical and Computational Framework for Reliable Real-Time Solution of Parametrized Partial Differential Equations

2002 ◽  
Vol 36 (5) ◽  
pp. 747-771 ◽  
Author(s):  
Christophe Prud'homme ◽  
Dimitrios V. Rovas ◽  
Karen Veroy ◽  
Anthony T. Patera
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Real Time ◽  
Partial Differential ◽  
Computational Framework ◽  
Time Solution ◽  
Parametrized Partial Differential Equations

Computations of Higher Class Solutions of Partial Differential Equations

2001 ◽  
Author(s):  
K. S. Surana ◽  
M. A. Bona
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Convergence Rates ◽  
Integral Form ◽  
Mathematical Framework ◽  
Partial Differential ◽  
Computational Framework ◽  
Fundamental Point ◽  
Adaptive Processes ◽  
Fixed Order

Abstract This paper presents a new computational strategy, computational framework and mathematical framework for numerical computations of higher class solutions of differential and partial differential equations. The approach presented here utilizes ‘strong forms’ of the governing differential equations (GDE’s) and least squares approach in constructing the integral form. The conventional, or currently used, approaches seek the convergence of a solution in a fixed (order) space by h, p or hp-adaptive processes. The fundamental point of departure in the proposed approach is that we seek convergence of the computed solution by changing the orders of the spaces of the basis functions. With this approach convergence rates much higher than those from h,p–processes are achievable and the progressively computed solutions converge to the ‘strong’ i.e. ‘theoretical’ solutions of the GDE’s. Many other benefits of this approach are discussed and demonstrated. Stationary and time-dependant convection-diffusion and Burgers equations are used as model problems.

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