Abstract
We consider the Dirichlet problem for an elliptic equation with a singularity. The singularity of the solution to the problem is caused by the presence of a re-entrant corner at the boundary of the domain. We define an Rν-generalized solution for this problem. This allows for the construction of numerical methods for finding an approximate solution without loss of accuracy. In this paper, the existence and uniqueness of the Rν-generalized solution in set W∘2,α1(Ω,δ) is proven. The Rν-generalized solution is the same for different parameters ν.
Highlights
IntroductionDirichlet–Dirichlet and Dirichlet–Neumann boundary conditions posed on the adjacent sides of re-entrant corners, it was stated that 0.25 ≤ k i ≤ 0.67 for 3π/2 ≤ γi ≤ 2π
For the boundary-value problem for elliptic equations with singularity, we propose to define its solutions as an Rν -generalized one
We presented the concept of an Rν -generalized solution for boundary-value problems with a singularity
Summary
Dirichlet–Dirichlet and Dirichlet–Neumann boundary conditions posed on the adjacent sides of re-entrant corners, it was stated that 0.25 ≤ k i ≤ 0.67 for 3π/2 ≤ γi ≤ 2π In this case, the approximate finite-element solution converges to a weak solution of the problem at an O(hk ) rate (h is the mesh step) in the norm of Sobolev space W21 (Ω) [1,2]. For the boundary-value problem for elliptic equations with singularity, we propose to define its solutions as an Rν -generalized one (see, for examples, [13]) This conception allows for investigating problems with singularities of different kinds (discontinuous or not bounded coefficients, right hands of the equation and boundary conditions; existence of the re-entrant corners on the boundary) and constructing the weighted FEMfor these problems. This restrictive assumption is dropped and a weighted set of functions is defined in which a unique Rν -generalized solution exists for the given class of the boundary value problems; this solution is unique for all ν
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
