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A fast Poisson solver of arbitrary order accuracy in rectangular regions
 SIAM J. Sci. Comput
, 1998
"... 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 largescale problems. The method is based on a pseudospectral Fourier approximation an ..."
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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 largescale 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.
Solution of Time Dependent Diffusion Equations with Variable Coefficients using Multiwavelets
"... A new numerical algorithm is developed for the solution of time dependent differential equations of the diffusion type. It allows for an accurate and efficient treatment of multidimensional problems with variable coefficients, nonlinearities and general boundary conditions. For space discretization ..."
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A new numerical algorithm is developed for the solution of time dependent differential equations of the diffusion type. It allows for an accurate and efficient treatment of multidimensional problems with variable coefficients, nonlinearities and general boundary conditions. For space discretization we use the multiwavelet bases introduced in [1] which were then applied to the representation of differential operators and functions of operators in [3]. An important advantage of multiwavelet basis functions, is the fact that they are supported only on non overlapping subdomains . Thus multiwavelet bases are attractive for solving problems in finite (non periodic) domains. Boundary conditions are imposed with a penalty technique of [12], it can be used to impose rather general boundary conditions. The penalty approach was extended to a procedure for ensuring the continuity of the solution and its first derivative across interior boundaries between neighboring subdomains while time stepping...
Asymptotic Fourier Coefficients for a C ∞ Bell (Smoothed“TopHat”) & the Fourier Extension Problem
, 2004
"... In constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonperiodic function, defined on an interval x ∈ [−χ,χ], to a function f ..."
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In constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonperiodic function, defined on an interval x ∈ [−χ,χ], to a function f ̃ which is periodic on the larger interval x ∈ [−Θ,Θ]. We derive the asymptotic Fourier coefficients for an infinitely differentiable function which is one on an interval x ∈ [−χ,χ], identically zero for x>Θ, and varies smoothly in between. Such smoothed “tophat ” functions are “bells ” in wavelet theory. Our bell is (for x≥ 0) T (x;L,χ,Θ) = (1 + erf(z))/2 where z = Lξ/ 1 − ξ2 where ξ ≡ −1 + 2(Θ − x)/(Θ − χ). By applying steepest descents to approximate the coefficient integrals in the limit of large degree j, we show that when the width L is fixed, the Fourier cosine coefficients aj of T on x ∈ [−Θ,Θ] are proportional to aj ∼ (1/j) exp(−Lπ1/22−1/2(1 − χ/Θ)1/2j1/2)Λ(j) where Λ(j) is an oscillatory factor of degree given in the text. We also show that to minimize error in a Fourier series truncated after the N th term, the width should be chosen to increase with N as L = 0.91√1−χ/ΘN1/2. We derive similar asymptotics for the function f (x) = x as extended by a more sophisticated scheme with overlapping bells; this gives an even faster rate of Fourier convergence.
JOURNAL OF COMPUTATIONAL PHYSICS 144, 109–136 (1998) ARTICLE NO. CP986001 A Fast 3D Poisson Solver of Arbitrary Order Accuracy
, 1997
"... We present a direct solver for the Poisson and Laplace equations in a 3D rectangular box. The method is based on the application of the discrete Fourier transform accompanied by a subtraction technique which allows reducing the errors associated with the Gibbs phenomenon and achieving any prescribed ..."
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We present a direct solver for the Poisson and Laplace equations in a 3D rectangular box. The method is based on the application of the discrete Fourier transform accompanied by a subtraction technique which allows reducing the errors associated with the Gibbs phenomenon and achieving any prescribed rate of convergence. The algorithm requires O(N 3 log N) operations, where N is the number of grid points in each direction. We show that our approach allows accurate treatment of singular cases which arise when the boundary function is discontinuous or incompatible with the differential equation. c ○ 1998 Academic Press Key Words: 3D Poisson solver for Dirichlet problem; Fourier method; corner and edge singularities.