We present boundary integral representations of several initial boundary value problems related to the heat equation. A Galerkin discretization with piecewise constant functions in time and piecewise linear functions in space leads to optimal a priori error estimates, provided that the meshwidths in space and time satisfy ht = O(h 2 x ). Each time step involves the solution of a linear system, whose spectral condition number is independent of the refinement under the same assumption on the mesh. We show that if the parabolic multipole method is used to apply parabolic boundary integral operators, the overall complexity of the scheme is log-linear while preserving the convergence of the Galerkin discretization method. The theoretical estimates are confirmed numerically at the end of the paper. [1,9], and Costabel [4]. For classically used second kind integral equations, e.g., a double-layer potential approach for the Dirichlet problem and a single-layer potential ansatz for the Neumann problem, the compactness of these integral operators for smooth domains guarantees the well-posedness and provides the backbone for the analysis of numerical methods. However, in the case of nonsmooth domains and first kind integral equations the situation is more complicated. Brown [3] gave some first results on Lipschitz domains before Arnold and Noon [1] and Costabel [4] proved almost contemporaneously the boundedness and coercivity of the thermal single-layer operator. Furthermore, the latter paper showed the coercivity of the hypersingular operator and the boundedness of all thermal boundary integral operators in the appropriate anisotropic Sobolev space setting.
Introduction. Boundary integral equations related to the heat equation have been studied in Pogorzelski [11], Brown [3], Arnold and NoonWith these results the analysis of Galerkin methods in space and time follows the well-known pattern of the elliptic theory, i.e., the Lax-Milgram lemma guarantees uniqueness and solvability of the corresponding operator equations and their Galerkin variational formulation. Using conforming finite-dimensional subspaces of the natural energy spaces, uniqueness and solvability translates directly to the discrete system, where Cea's lemma provides quasi-optimality. Using the approximation property of piecewise polynomial finite-dimensional ansatz spaces, the regularity of the boundary integral operators, and assuming certain regularity of the discretization, one can derive explicit error estimates in the energy norm as well as weaker and stronger norms.