On the Existence and Uniqueness of an Rν-Generalized Solution to the Stokes Problem with Corner Singularity

Abstract

We consider the Stokes problem with the homogeneous Dirichlet boundary condition in a polygonal domain with one re-entrant corner on its boundary. We define an Rν-generalized solution of the problem in a nonsymmetric variational formulation. Such defined solution allows us to construct numerical methods for finding an approximate solution without loss of accuracy. In the paper, the existence and uniqueness of an Rν-generalized solution in weighted sets is proved.

On the Error Estimation of the FEM for the Nikol’skij-Lizorkin Problem with Degeneracy in the Lebesgue Space

“…In the future, we plan to define an R ν -generalized solution (see [31][32][33]) for the Nikol'skii-Lizorkin problem. This will make it possible to achieve the convergence rate of the solution by the finite element method equal to O(h 2 ) without compression of the mesh to the boundary.…”

The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with Singularity

“…This makes it possible to construct numerical methods of a high order of accuracy, i.e., with the rate of convergence of the approximate solution to the exact one O(h 2 ). Existence and uniqueness, coercivity and differential properties of the solution in weighted Sobolev spaces were studied in [21][22][23][24][25]. A weighted finite element method for finding an approximate R ν -generalized solution with a convergence rate of O(h) was developed in [26].…”