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A space‐domain preconditioned spectral method for non‐linear boundary value problems
Author(s) -
Axelrad V.
Publication year - 1992
Publication title -
international journal of numerical modelling: electronic networks, devices and fields
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.249
H-Index - 30
eISSN - 1099-1204
pISSN - 0894-3370
DOI - 10.1002/jnm.1660050202
Subject(s) - boundary value problem , mathematics , spectral method , weighting , galerkin method , rate of convergence , iterative method , linear system , eigenvalues and eigenvectors , operator (biology) , partial differential equation , numerical analysis , convergence (economics) , mathematical analysis , algorithm , mathematical optimization , nonlinear system , computer science , medicine , computer network , channel (broadcasting) , biochemistry , physics , chemistry , quantum mechanics , repressor , transcription factor , gene , economics , radiology , economic growth
A novel spectral procedure for the numerical solution of non‐linear boundary value problems is presented. The discrete spectral equations are solved by an iterative algorithm using space‐domain preconditioning. The preconditioning operator is obtained by spatial weighting (or windowing) of the exact differential operator. Convergence behaviour of the iterative solution is investigated using an eigenvalue analysis. Theoretical estimates for the convergence rate and accuracy are compared with that achieved in a numerical application—a non‐linear boundary value problem from semiconductor device modelling. The method combines the infinite‐order exponential accuracy of spectral discretizations with the sparse structure of finite difference equations. This offers numerical advantages in comparison to previously developed Fourier–Galerkin algorithms, particularly important for physically ill‐conditioned and strongly non‐linear problems. The tradeoff of achievable accuracy versus computer time is easily controlled, thus making essential speed‐ups possible for moderate accuracy requirements.