On the Spline Collocation Method for the Single-layer Heat Operator Equation

We consider a boundary element collocation method for the heat equation. As trial functions we use the tensor products of continuous piecewise linear splines with collocation at the nodal points. Convergence and stability is proved in the case where the spatial domain is a disc. Moreover, practical implementation is discussed in some detail. Numerical experiments confirm our results. Introduction Recently, the Boundary Element Method has been applied to the solution of various time-dependent phenomena such as heat conduction governed by homogeneous parabolic equations [7, 8, 15, 16, 19, 20] and wave propagation governed by hyperbolic equations [5, 6]. The BEM solution of time-dependent problems requires a large computational effort, since results from all previous steps are saved in the computer memory. In addition, careful numerical integration has to be carried out for setting up the matrix equations, which means increasing computing time. Thus, there is a need to reduce the effort, for example by applying simple discretization methods such as collocation or the quadrature method instead of the Galerkin approximation. In the above-mentioned articles [7, 19] only the Galerkin solution is analyzed. The work [20] of Onishi describes a collocation scheme for the heat equation when the boundary integral equation is of the second kind. On the other hand, there are well-known situations which lead to boundary integral equations of the first kind. In such time-dependent cases no results for the collocation method seem to be available. Therefore, we are inevitably faced with the question of effectively solving such equations. For time-independent problems the collocation, and more recently the quadrature methods, have been analyzed [23, 24]. In this paper we construct and analyze a spline collocation scheme for solving the single-layer heat operator equation, assuming that the spatial domain is two-dimensional and has a smooth boundary. As trial functions we use tensor products of continuous piecewise linear functions with collocation at the nodal points. For the proof of stability and convergence we limit ourselves here to the case of the circle. However, the method can be applied to all smooth Received by the editor February 27, 1992. 1991 Mathematics Subject Classification. Primary 65N35, 65R20, 45L10, 35K05.