together several new ideas and compare them with more classical techniques. There are several fundamental ingre-This paper presents a new fictitious domain formulation for the solution of a strongly elliptic boundary value problem with Neu-dients coming together here: mann boundary conditions for a bounded domain in a finite-dimen-β’ The boundary value problem is formulated for an open sional Euclidean space with a smooth (possibly only Lipschitz)
domain with a rectifiable boundary of any shape.
boundary. This extends the domain to a larger rectangular domain with periodic boundary conditions for which fast solvers are avail-
β’ The given domain is embedded in a larger and simpler able. The extended solution converges on the original domain in domain (usually rectilinear in shape).
the appropriate function spaces as the penalty parameter approaches zero. Both wavelet-Galerkin and finite elements numerical
β’ The elliptic boundary-value problem in the original approximation schemes are developed using this methodology. The domain is reformulated in a weak form as an integral equaconvergence rates of both schemes are comparable, and the use tion in the larger domain, and this involves introducing a of finite elements requires a parameterization of the boundary, while regularization parameter (the so-called penality paramethe wavelet-Galerkin method admits an implicit description of the ter). Solutions depending on converge to solutions of the boundary in terms of a wavelet representation of the boundary measure defined as the distributional gradient of the characteristic original equation as converges to zero.
function of the interior. The accuracy of both methods is investigated
β’ Both wavelet and finite-element Galerkin type methand compared, both theoretically and for numerical test cases. The ods are used for numerical approximations in the larger conclusion is that the methods are comparable, and that the wavelet domain for fixed and small values of . method allows the use of more general boundaries which are not explicitly parametrized, which would be of greater advantage in β’ Due to the rectinlinear nature of the larger domain, higher dimensions (the numerical tests are carried out in two fast periodic solvers can be implemented in the larger dimensions).