Abstract

The finite element method has established itself as an efficient numerical procedure for the solution of arbitrary-shaped field problems in space. Basically, the finite element method transforms the underlying differential equation into a system of algebraic equations by application of the method of weighted residuals in conjunction with a finite element ansatz. However, this procedure is restricted to even-ordered differential equations and leads to symmetric system matrices as a key property of the finite element method. This paper aims in a generalization of the finite element method towards the solution of first-order differential equations. This is achieved by an approach which replaces the first-order derivative by fractional powers of operators making use of the square root of a Sturm-Liouville operator. The resulting procedure incorporates a finite element formulation and leads to a symmetric but dense system matrix. Finally, the scheme is applied to the barometric equation where the results are compared with the analytical solution and other numerical approaches. It turns out that the resulting numerical scheme shows excellent convergence properties.

Highlights

  • Many problems in physics are described by differential equations which in general can only be solved numerically

  • This paper aims in a generalization of the finite element method towards the solution of first-order differential equations

  • The overall goal is to establish a method that can be applied to any spatial first-order differential equation which results in conjunction with a finite element ansatz in symmetric system matrices

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Summary

Introduction

Many problems in physics are described by differential equations which in general can only be solved numerically. Due to the occurrence of left and right fractional derivatives the resulting system matrix was non-symmetric and he failed to succeed [1] For this reason, in the following a different approach is applied which makes use of fractional powers of operators [2] [3] [4]. The overall goal is to establish a method that can be applied to any spatial first-order differential equation which results in conjunction with a finite element ansatz in symmetric system matrices.

Schmidt et al DOI
Operator Equation
Derivation of the Finite Element Formulation
Numerical Implementation
Conclusion
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