Implicit discretization of time dependent problems in computational physics, semiconductor device simulation, electromigration and fluid dynamics often gives rise to elliptic equations. Helmholtz type equations usually appear in acoustics or electromagnetics and also as a result of time discretization of the Navier-Stokes equations. Biharmonic problems appear in elasticity and viscous flows.
We solve the Helmholtz and the biharmonic equation by the Domain Decomposition (DD) methods. Previously  we adopted a DD method where the equation was solved in each subdomain with assumed boundary conditions, resulting in jumps in function or derivative on subdomain boundaries. These jumps were removed by the introduction of singularity layers. In order to account for the global effect of the layers we computed first the influence of each layer on each subdomain boundary, taking into account the decay or smoothing out of the influence as a function of the distance from the layer. To reduce the communication load, compression in a multiwavelet basis was applied. Nevertheless, this part of the procedure can become expensive as the number of subdomains grows considerably. An algorithm for a fast solution of the Poisson equation by decomposition of the domain into square domains and the subsequent matching of these solutions by the fast multipole method was developed in .
The algorithm developed in  incorporates the following steps:
1. In each subdomain a particular solution of the non-homogeneous equation with arbitrary Neumann (Dirichlet) boundary conditions is found.
2. The collection of particular solutions usually has discontinuities (or discontinuities in the derivatives) on the boundaries of the subdomains. We introduce double (single) layers on the boundaries to match the solutions from different domains to have continuous global solution. The effect of these layers on other boundaries is calculated.
3. The solutions obtained at the first step are patched by adding the solutions of the Laplace equation with the boundary conditions that were computed in the previous step.
4. An additional solution of the corresponding homogeneous equation is added to satisfy the global boundary conditions.
Considerable improvement in the efficiency of
the interface jump removal step can be achieved if at each step only
adjacent boxes are matched. This is the basis of the hierarchical
approach which was proposed in .
The "elementary step" of the hierarchical algorithm is the following.
1. First, for each two adjacent subdomains some boundary conditions are defined. These conditions should not contradict the given right hand side at the junctions (see ). The Poisson equation is solved with these boundary conditions by a fast spectral algorithm .
2. The solutions have a discontinuity in the first derivative. We match the subdomains by adding certain discontinuous functions. In fact we only evaluate these functions at the boundaries of two adjacent subdomains and then solve the homogeneous equation in each subdomain with the cumulative boundary conditions.
3. The global homogeneous equation is solved in such a way that it satisfies the assumed conditions at the ``global boundaries" of the merged subdomains. The solution of the non-homogeneous equation is expensive if compared to the homogeneous equation where efficient algorithms are available .
This step is repeated hierarchically. For example, first the smallest adjacent domains are matched: box 1 with box 2, box 3 with box 4, then the merged box 1,2 is matched with 3,4, afterwards the box which is a union of boxes 1,2,3,4 is patched with the adjacent merged box etc.
Here we extend the results of [1,4] in the following directions.
1. In addition to the Poisson equation, we solve the Helmholtz or modified Helmholtz equation. One of the problems that arise is resonance and non-resonance situations for Dirichlet and Neumann boundary conditions. This can be avoided by introducing "transparent" boundary conditions which are approximations to the Sommerfeld radiation boundary condition. This means that the scattered wave behaves asymptotically like a diverging spherical wave. Some modifications of the non-reflecting conditions were considered in .
2. We solve the biharmonic equation with either a free boundary as in  or boundary conditions of one type only (Dirichlet or Neumann). The biharmonic equation is split into a coupled system of the Poisson equations. The unknown function satisfies the Poisson equation with a certain right hand side which will be called a vorticity function. The solution is also performed by DD. Here the matching algorithm which was developed in [1,4] is applied to find the smooth vorticity function which matches the given right hand side. Then we obtain a solution of the biharmonic equation. The global boundary conditions of either Dirichlet or Neumann type can be satisfied by the addition of an appropriate harmonic function, as was done for the solution of the Poisson equation.
1. A. Averbuch, E. Braverman and M. Israeli,
Parallel adaptive solution of
a Poisson equation with multiwavelets,
SIAM J. Sci.Comput. 22(2000), 1053-1086.
2. L. Greengard and J.-Y. Lee, A direct adaptive Poisson solver of arbitrary order accuracy, J. Comput. Phys. 125 (1996), 415-424.
3. A. Averbuch, M. Israeli, L. Vozovoi, A fast Poisson solver of arbitrary order accuracy in rectangular regions, SIAM J. of Sci. Comput. 19(1998), 933-952.
4. M. Israeli, E. Braverman and A. Averbuch, A Hierarchical domain decomposition method with low communication overhead, submitted to Proceedings of the 13th Domain Decomposition Conference, Lyon, 2000.
5. A. Bamberger, P. Joly and J.E. Roberts, Second-order absorbing boundary conditions for the wave equation: a solution for the corner, SIAM J. Numer. Anal. 27 (1990), 323-352.