ISSN:
1572-9125
Keywords:
65L06
;
65M20
;
65M60
;
singly implicit Runge-Kutta methods
;
finite-element methods
;
adaptive mesh refinement
;
error estimation
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract We describe an adaptive mesh refinement finite element method-of-lines procedure for solving one-dimensional parabolic partial differential equations. Solutions are calculated using Galerkin's method with a piecewise hierarchical polynomial basis in space and singly implicit Runge-Kutta (SIRK) methods in time. A modified SIRK formulation eliminates a linear systems solution that is required by the traditional SIRK formulation and leads to a new reduced-order interpolation formula. Stability and temporal error estimation techniques allow acceptance of approximate solutions at intermediate stages, yielding increased efficiency when solving partial differential equations. A priori energy estimates of the local discretization error are obtained for a nonlinear scalar problem. A posteriori estimates of local spatial discretization errors, obtained by order variation, are used with the a priori error estimates to control the adaptive mesh refinement strategy. Computational results suggest convergence of the a posteriori error estimate to the exact discretization error and verify the utility of the adaptive technique.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01989753
Permalink
