scholarly journals Convergence of the Finite Element Method for Boundary Value Problem with Degeneration on Whole Boundary of the Domain

2019 ◽  
Vol 8 (3) ◽  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
The Finite Element Method ◽  
Element Method

Stable finite element method for solving the oblique derivative boundary value problems in geodesy

2021 ◽  
Author(s):  
Marek Macák ◽  
Zuzana Minarechová ◽  
Róbert Čunderlík ◽  
Karol Mikula
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
The Finite Element Method ◽  
Gravity Field Modelling ◽  
Field Modelling ◽  
Local Gravity ◽  
Local Gravity Field Modelling ◽  
Element Method

<p><span>We presents local gravity field modelling in a spatial domain using the finite element method (FEM). FEM as a numerical method is applied for solving the geodetic boundary value problem with oblique derivative boundary conditions (BC). We derive a novel FEM numerical scheme which is the second order accurate and more stable than the previous one published in [1]. A main difference is in applying the oblique derivative BC. While in the previous FEM approach it is considered as an average value on the bottom side of finite elements, the novel FEM approach is based on the oblique derivative BC considered in relevant computational nodes. Such an approach should reduce a loss of accuracy due to averaging. Numerical experiments present </span><span>(i) </span><span>a reconstruction of EGM2008 as a harmonic function over the extremely complicated Earth’s topography in the Himalayas and Tibetan Plateau, and (ii) local gravity field modelling in Slovakia with the high-resolution 100 x 100 m while using terrestrial gravimetric data.</span></p><p><span>[1] </span>Macák, Z. Minarechová, R. Čunderlík, K. Mikula, The finite element method as a tool to solve the oblique derivative boundary value problem in geodesy. Tatra Mountains Mathematical Publications. Vol. 75, no. 1, 63-80, (2020)</p>

Convergence of the Finite Element Method in the Theory of Elasticity

1968 ◽  
Vol 35 (2) ◽  
pp. 274-278 ◽  
Author(s):  
M. W. Johnson ◽  
R. W. McLay
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Variational Principle ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Careful Consideration ◽  
Stress Singularities ◽  
The Finite Element Method ◽  
Element Method

The foundations of the theory of the finite element method as it applies to linear elasticity are investigated. A particular boundary-value problem in plane stress is considered and the variational principle for the finite element method is shown to be equivalent to it. Mean and uniform convergence of the finite element solution to that of the boundary-value problem is demonstrated with careful consideration given to the stress singularities. A counterexample is presented in which a set of functions, admissible to the variational principle, is shown not to converge.

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