Stability of a spline collocation method for strongly elliptic multidimensional singular integral equations

Abstract

We consider a spline collocation method for strongly elliptic zero order pseudodifferential equations peAu = f on a cube f2 = (0, l) m. Utilizing multilinear spline functions which are zero at the boundary ~?f2 we collocate at the meshpoints inside ~2. For classical strongly elliptic translation invariant pseudodifferential operators, we verify the stability of the considered collocation method in L2(Q). Afterwards, for m < 2 and a right hand side .f~HS(Q), s > m/2, we prove an asymptotic convergence estimate.

Convergence of boundary element collocation methods for dirichlet and neumann screen problems in R3

“…Here the functions (numerical symbols) XG, XC,XWIG and AwlC are defined by Proof: The proof of the equations (3.14a,b) was done in [7]. For more detailed computations see [24]. We give the necessary formulas for the derivation of equations (3.14qd).…”

“…Then the symbolic calculus for Toeplitz matrices can be applied. For the case of pseudodifferential operators of order zero on 0 this method was used in [24] by Schneider. In [7] we derived results for V, an pseudodifferential operator of order -1 using trial functions of odd degree.…”

Spline collocation for strongly elliptic equations on the torus

“…For a translation invariant operator on an interval, the collocation matrices are Toeplitz matrices, and the stability analysis of the collocation method can be reduced to that of the finite section method for an infinite Toeplitz matrix. This approach has recently been generalized to collocation methods on the cube [0, 1]" for translation invariant singular integral operators [26], for translation invariant operators of order -1, including the practically important case of the first kind integral equation with the single layer potential operator [10], and to arbitrary strongly elliptic pseudodifferential operators of order zero, also on more general domains and the torus [23].…”

Gemischte Fourier‐Randelementmethoden zur Berechnung zeitharmonischer Streufelder an Schirmen

“…Untersuchungen uber die Konvergenz von Kollokationsverfahren auf zweidimensionalen Mannigfaltigkeiten sind neueren Datums [14,15,16,55,56,61] und noch nicht abgeschlossen.…”