Approximation methods for the Muskhelishvili equation on smooth curves

Abstract

Abstract. We investigate the possibility of applying approximation methods to the famous Muskhelishvili equation on a simple closed smooth curve Γ. Since the corresponding integral operator is not invertible the initial equation has to be corrected in a special way. It is shown that the spline Galerkin, spline collocation and spline qualocation methods for the corrected equation are stable, and the corresponding approximate solutions converge to an exact solution of the Muskhelishvili equation in appropriate norms. Numerical experiments confirm the effectiveness of the proposed methods.

On the Stability of the Nyström Method for the Muskhelishvili Equation on Contours with Corners

“…Note that the kernels k(t, τ ) of the integral operators K Γ and L Γ possess finite limits lim t→τ k(t, τ ) for any τ / ∈ M Γ [10]. This allows us to define the expression k(τ, τ ) by…”

“…A systematic study of such problems have been started only recently. Thus in the space L 2 (Γ) the stability of spline Galerkin, spline collocation, and qualocation methods on simple smooth curves has been established in [10]. For such contours, the methods mentioned are always stable.…”

Stability of the Nyström Method for the Sherman–Lauricella Equation

“…Note that if k(t, τ ) is the kernel of the integral operator L Γ or K Γ , then for any point τ ∈ Γ \ M Γ , there is a finite limit lim t→τ k(t, τ ) [8]. Therefore, for any…”

“…Recall that localizing principles connect the invertibility of an element from a given Banach or C * -algebra with the invertibility of its local representatives [1,12]. Notice that in the neighborhoods of nonangular points τ ∈ Γ, τ / ∈ M Γ , the integral operators of (1.4) behave like compact operators [8], so for such points τ ∈ Γ our approximation sequence (A n ) n∈N is locally equivalent to the sequence of projections (P n ) n∈N which is a stable sequence. Thus we need only identify and study local representatives of the Nyström method for the angular points τ j ∈ Γ.…”

Critical Angles of the Nyström Method for Double Layer Potential Equation

“…Let k = k(t, τ) denote the kernel of the double layer potential operator V Γ . Straightforward calculations [16] show that for any τ /…”