On the Error Estimation of the Finite Element Method for the Boundary Value Problems with Singularity in the Lebesgue Weighted Space

On the Existence and Uniqueness ofRv-Generalized Solution for Dirichlet Problem with Singularity on All Boundary

“…2,]+ /2 (Ω). In fact, by virtue of conditions (9), (12), and (15) and the Cauchy-Schwarz inequality, we have…”

“…For boundary value problems for elliptic equations, Maxwell equations and Lamé system, we constructed the numerical methods with rate of convergence independent of the singularity based on the concept of an ] -generalized solution (see, e.g., [10][11][12]).…”

“…Let conditions (9)-(12) and (15) hold and let Then, the ] -generalized solution ] of the Dirichlet problem with coordinated degeneration of the input data on all boundary of the domain exists and is unique in the space∘ 1…”

The Properties of the Weighted Space Hk 2,α (Ω) and Weighted Set Wk 2,α(Ω, δ)

“…Such a new concept of solution led to the distinction of two classes of boundary value problems: problems with coordinated and uncoordinated degeneracy of input data; it also made it possible to study the existence and uniqueness of solutions as well as its coercivity and differential properties in the weighted spaces and weighted sets (see [18]- [21]). In [22]- [26], we constructed and investigated the finite-element method for different boundary value problems.…”

Weighted Finite-Element Method for Elasticity Problems with Singularity

“…The existence and uniqueness of solutions as well as its coercivity and differential properties in the weighted Sobolev spaces and sets were proved [5][6][7][8][9][10], the weighted finite-element method was built, and its convergence rate was investigated [11][12][13][14][15].…”