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Abstract. In this paper we propose a direct method for the solution of the Poisson equation in rectangular regions. It has an arbitrary order accuracy and low CPU requirements which makes it practical for treating large-scale problems.The method is based on a pseudospectral Fourier approximation and a polynomial subtraction technique. Fast convergence of the Fourier series is achieved by removing the discontinuities at the corner points using polynomial subtraction functions. These functions have the same discontinuities at the corner points as the sought solution. In addition to this, they satisfy the Laplace equation so that the subtraction procedure does not generate nonperiodic, nonhomogeneous terms.The solution of a boundary value problem is obtained in a series form in O(N log N ) floating point operations, where N 2 is the number of grid nodes. Evaluating the solution at all N 2 interior points requires O(N 2 log N ) operations.Key words. boundary value problem, Poisson equation, rectangular region, spectral method, corner discontinuities, polynomial subtraction AMS subject classifications. 35J05, 45L10, 65P05PII. S1064827595288589 1.Introduction. An important step in the development of fast numerical solvers for elliptic equations in complicated domains is an algorithm for the solution of boundary value problems for constant coefficient elliptic equations in rectangular regions. After solving a nonhomogeneous equation with some "convenient" boundary conditions, one must solve, in a correction step, the homogeneous problem with specified (Dirichlet or Neumann) boundary conditions. As a resulting boundary distribution may not satisfy the required compatibility conditions at the corner points, singularities may arise in the process of the solution, thus destroying the convergence rate even if the final solution is smooth.When the computational region is discretized on a N × N grid using some loworder (finite difference or finite element) scheme, the resulting system of linear algebraic equations is represented by a sparse matrix. Such a matrix can be inverted in O(N 2 ) (or O(N 2 log 2 N ) arithmetic operations using a "fast solver" [4]. However, since the method is of low order, the resolution N must be very large if a high accuracy is desired.Application of high-order (pseudo) spectral methods, based on global expansions into orthogonal polynomials, e.g., Chebyshev polynomials, to the solution of elliptic equations, results in a full matrix problem. The cost of inverting such a matrix using the best current algorithms is O(N 3 ) operations [2]. Besides, the accuracy decreases considerably as the dimension N 2 of the system grows due to accumulation of roundoff errors.When using the Chebyshev method, the fast computation of the expansion coefficients requires that the problem be discretized on a nonuniform grid. For timedependent problems, when elliptic equations arise due to a time-discretization pro-

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It is well known that the Fourier partial sum of an analytic nonperiodic function, supported on a finite interval, converges slowly inside the interval and exhibits O(1) spurious oscillations near the boundaries (the Gibbs phenomenon). An effective algorithm which allows one to completely overcome the Gibbs phenomenon was developed in [J. Comput. Appl. Math., 43 (1992), pp. 81-92]. The basic concept of this approach consists of the re-expansion of the Fourier partial sums into the rapidly convergent Gegenbauer series.In this paper we extend the Fourier-Gegenbauer (F-G) method of [J. Comput. Appl. Math., 43 (1992), pp. 81-92] to the evaluation of the spatial derivatives of a piecewise analytic function. Also, we apply this method to the solution of nonperiodic boundary value problems. Although the derivatives of a discontinuous function are not in L 2 , the exponential convergence of the truncated Gegenbauer series can be proved, and the rate of convergence can be estimated.The solution of differential equations is accomplished in two steps. First, a particular solution with arbitrary boundary conditions is constructed using the F-G method. This particular solution is then corrected to satisfy the prescribed boundary conditions of the problem by adding a proper linear combination of homogeneous solutions. For boundary layer problems the intermediate (particular) solution has steep profiles near the boundaries. These steep regions introduce a large error into the final solution, which presumably has a smooth profile on the whole interval. A method which compensates for this loss of accuracy by using the appropriately constructed homogeneous solutions is proposed.

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