A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and related Volterra-Fredholm integral equations with smooth solutions

2019 ◽  
Vol 84 (1) ◽  
pp. 63-89 ◽  
Author(s):  
Mahmoud A. Zaky ◽  
Ibrahem G. Ameen
Keyword(s):  
Nonlinear Systems ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Spectral Method ◽  
Error Estimates ◽  
A Priori ◽  
Fredholm Integral Equations ◽  
Smooth Solutions ◽  
Priori Error Estimates ◽  
Fractional Boundary Value Problems

scholarly journals A note on a priori error estimates for augmented mixed methods

2016 ◽  
Vol 57 ◽  
pp. 139-144
Author(s):  
Tomás P. Barrios ◽  
Edwin Behrens ◽  
Rommel Bustinza
Keyword(s):  
Mixed Methods ◽  
Error Estimates ◽  
A Priori ◽  
A Priori Error Estimates ◽  
Priori Error Estimates

scholarly journals Application of boundary integral equation method to numerical solution of elliptic boundary-value problems in R3

2020 ◽  
Vol 22 (3) ◽  
pp. 319-332
Author(s):  
Aleksandr N. Tynda ◽  
Konstantin A. Timoshenkov
Keyword(s):  
Boundary Value Problems ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Value ◽  
Spline Approximation ◽  
Boundary Integral ◽  
Double Layers ◽  
Fredholm Integral Equations ◽  
Fredholm Type ◽  
Graded Meshes

In this paper we propose numerical methods for solving interior and exterior boundary-value problems for the Helmholtz and Laplace equations in complex three-dimensional domains. The method is based on their reduction to boundary integral equations in R2. Using the potentials of the simple and double layers, we obtain boundary integral equations of the Fredholm type with respect to unknown density for Dirichlet and Neumann boundary value problems. As a result of applying integral equations along the boundary of the domain, the dimension of problems is reduced by one. In order to approximate solutions of the obtained weakly singular Fredholm integral equations we suggest general numerical method based on spline approximation of solutions and on the use of adaptive cubatures that take into account the singularities of the kernels. When constructing cubature formulas, essentially non-uniform graded meshes are constructed with grading exponent that depends on the smoothness of the input data. The effectiveness of the method is illustrated with some numerical experiments.

scholarly journals On Superconvergence of a Gradient for Finite Element Methods for an Elliptic Equation with the Nonsmooth Right–hand Side

2002 ◽  
Vol 2 (3) ◽  
pp. 295-321 ◽  
Author(s):  
Alexander Zlotnik
Keyword(s):  
Finite Elements ◽  
Elliptic Equation ◽  
Error Estimates ◽  
Dirichlet Boundary ◽  
A Priori ◽  
Boundary Function ◽  
Right Hand ◽  
Priori Error Estimates ◽  
The Right ◽  
2D And 3D

AbstractThe elliptic equation under the nonhomogeneous Dirichlet boundary condition in 2D and 3D cases is solved. A rectangular nonuniform partition of a domain and polylinear finite elements are taken. For the interpolant of the exact solution u, a priori error estimates are proved provided that u possesses a weakened smoothness. Next error estimates are in terms of data. An estimate is established for the right–hand side f of the equation having a generalized smoothness. Error estimates are derived in the case of f which is not compatible with the boundary function. The proofs are based on some propositions from the theory of functions. The corresponding lower error estimates are also included; they justify the sharpness of the estimates without the logarithmic multipliers. Finally, we prove similar results in the case of 2D linear finite elements and a uniform partition.

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