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Variational Problems in Domains with Cusp-Points and the Finite Element Method

ABSTRACT A variational problem in a two-dimensional domain with cusp-points corresponding to a linear elliptic boundary value problem is formulated and the unique existence of its solution is proved. The corresponding finite element method using triangular finite C 0-elements with polynomials of the first degree is analyzed and both the convergence (under the assumptions sufficient for the existence of the exact solution) and the maximal rate of convergence 𝒪(h) are proved.