Variational Methods for Numerical Solution of Boundary-Value Problems for Differential Equations. Finite Element Method. Boundary Element Method

This article considered the traditional finite element method (FEM) and adaptive finite element method (FEM) for the numerical solution of the one-dimensional boundary value problems. We established the preference or the superiority of the h-adaptive FEM to traditional FEM in high gradient problems in terms of accuracy and cost of computation. Numerical examples which confirm the performance and adaptability of the h-adaptive method over the traditional finite element method and the high accuracy of the numerical solution are presented. Detailed error analysis of linear elements was also discussed. In conclusion, h-adaptive FEM is recommended for complex systems with high gradient problems.

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The implementation of the boundary element method to the Helmholtz equation of acoustics

Information and Control Systems ◽

10.31799/1684-8853-2021-2-13-19 ◽

2021 ◽

pp. 13-19

Author(s):

Sergey Sivak ◽

Mihail Royak ◽

Ilya Stupakov ◽

Aleksandr Aleksashin ◽

Ekaterina Voznjuk

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Boundary Element Method ◽

Numerical Methods ◽

Helmholtz Equation ◽

Boundary Value Problems ◽

Boundary Element ◽

Three Dimensional ◽

The Finite Element Method ◽

Element Method

Introduction: To solve the Helmholtz equation is important for the branches of engineering that require the simulation of wave phenomenon. Numerical methods allow effectiveness’ enhancing of the related computations. Methods: To find a numerical solution of the Helmholtz equation one may apply the boundary element method. Only the surface mesh constructed for the boundary of the three-dimensional domain of interest must be supplied to make the computations possible. This method’s trait makes it possible toconduct numerical experiments in the regions which are external in relation to some Euclidian three-dimensional subdomain bounded in the three-dimensional space. The later also provides the opportunity of not using additional geometric techniques to consider the infinitely distant boundary. However, it’s only possible to use the boundary element methods either for the homogeneous domains or for the domains composed out of adjacent homogeneous subdomains. Results: The implementation of the boundary elementmethod was committed in the program complex named Quasar. The discrepancy between the analytic solution approximation and the numerical results computed through the boundary element method for internal and external boundary value problems was analyzed. The results computed via the finite element method for the model boundary value problems are also provided for the purpose of the comparative analysis done between these two approaches. Practical relevance: The method gives an opportunityto solve the Helmholtz equation in an unbounded region which is a significant advantage over the numerical methods requiring the volume discretization of computational domains in general and over the finite element method in particular. Discussion: It is planned to make a coupling of the two methods for the purpose of providing the opportunity to conduct the computations in the complex regions with unbounded homogeneous subdomain and subdomains with substantial inhomogeneity inside.

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