Finite Element Solution to the Helmholtz Equation with High Wave Number. Part 1. The h-Version of the FEM

The h-version of the finite element method with piecewise linear approximation has been applied to solve a one-dimensional model problem for the Helmholtz equation. The main practical purpose of the investigation is to lay theoretical ground for safe rules of the thumb how to choose the mesh width o...

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Bibliographic Details
Main Authors: Ihlenburg, Frank, Babuska, Ivo
Format: Report
Language:English
Subjects:
APPROXIMATION(MATHEMATICS)
BOUNDARY VALUE PROBLEMS
ERRORS
FEM(FINITE ELEMENT METHOD)
FINITE ELEMENT ANALYSIS
GREENS FUNCTIONS
HELMHOLTZ EQUATION
HILBERT SPACE
Numerical Mathematics
ONE DIMENSIONAL
QUASIOPTIMALITY
SOLUTIONS(GENERAL)
SPECIAL FUNCTIONS(MATHEMATICS)
STABILITY
Theoretical Mathematics
WAVE EQUATIONS
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Summary:The h-version of the finite element method with piecewise linear approximation has been applied to solve a one-dimensional model problem for the Helmholtz equation. The main practical purpose of the investigation is to lay theoretical ground for safe rules of the thumb how to choose the mesh width of the FE-model depending on the wavenumber. In this context we present new results for stability and error estimation of the FE-solution. Following the analysis, numerical results are discussed. In a second paper we will study the h-p-method for Helmholtz problems.