Abstract:A new method is introduced for solving Laplace problems on 2D regions with corners by approximation of boundary data by the real part of a rational function with fixed poles exponentially clustered near each corner. Greatly extending a result of D. J. Newman in 1964 in approximation theory, we first prove that such approximations can achieve root-exponential convergence for a wide range of problems, all the way up to the corner singularities. We then develop a numerical method to compute approximations via lin… Show more
“…We finish this section with a look at lightning solvers for PDEs in two-dimensional domains, introduced in 2019 and applied to date to Laplace [ 15 , 16 , 59 ], Helmholtz [ 16 ], and biharmonic equations (Stokes flow) [ 6 ]. In the basic case of a Laplace problem , the idea is to represent the solution on a domain E as , the real part of a rational function with no poles in E that approximates the boundary data to an accuracy typically of 6–10 digits.…”
Section: Root-exponential Convergence and Exponential Clustering Of Pmentioning
confidence: 99%
“… Fig. 4 Example of the lightning Laplace solver [ 15 , 16 ] as implemented in the code laplace.m [ 59 ]. For each number of degrees of freedom (DoF), poles are clustered exponentially near the 12 corners of the domain E , and the numbers are increased until a solution to 10-digit accuracy is obtained in the form of a rational function with 480 poles.…”
Section: Root-exponential Convergence and Exponential Clustering Of Pmentioning
confidence: 99%
“…We have not presented data in this section for lightning PDE solutions, but it was in this context that we first became aware of the importance of tapered exponential clustering. In the course of the work leading to [ 15 ], the first author noticed that although straight exponential spacing of preassigned poles gave root-exponential convergence, better efficiency could be achieved if the resulting approximations were re-approximated a second time by the AAA algorithm. On examination it was found that the AAA approximations had poles in a tapered distribution, just like cases A–C of Fig.…”
“…We finish this section with a look at lightning solvers for PDEs in two-dimensional domains, introduced in 2019 and applied to date to Laplace [ 15 , 16 , 59 ], Helmholtz [ 16 ], and biharmonic equations (Stokes flow) [ 6 ]. In the basic case of a Laplace problem , the idea is to represent the solution on a domain E as , the real part of a rational function with no poles in E that approximates the boundary data to an accuracy typically of 6–10 digits.…”
Section: Root-exponential Convergence and Exponential Clustering Of Pmentioning
confidence: 99%
“… Fig. 4 Example of the lightning Laplace solver [ 15 , 16 ] as implemented in the code laplace.m [ 59 ]. For each number of degrees of freedom (DoF), poles are clustered exponentially near the 12 corners of the domain E , and the numbers are increased until a solution to 10-digit accuracy is obtained in the form of a rational function with 480 poles.…”
Section: Root-exponential Convergence and Exponential Clustering Of Pmentioning
confidence: 99%
“…We have not presented data in this section for lightning PDE solutions, but it was in this context that we first became aware of the importance of tapered exponential clustering. In the course of the work leading to [ 15 ], the first author noticed that although straight exponential spacing of preassigned poles gave root-exponential convergence, better efficiency could be achieved if the resulting approximations were re-approximated a second time by the AAA algorithm. On examination it was found that the AAA approximations had poles in a tapered distribution, just like cases A–C of Fig.…”
“…Another is for the fast solution of special cases of the Laplace equation that are especially amenable to solution using complex-variable methods (Gopal & Trefethen, 2019).…”
Complex variables offer a unique and powerful way to represent planar regions and maps between them. Since the work of Riemann and others in the 19th century, complex variables have been used to derive quantities and properties of importance to a wide range of applications in science and engineering, in addition to their inherent mathematical importance. The advent of computing accelerated and extended this trend These packages tend to take the traditional form of library routines that require some degree of facility with custom data structures used to represent geometry and functions. Moreover, computational methods are mostly specialized to particular types or classes of regions, leading to parallel and duplicated functionality within a package, and incompatible representations between them.The ComplexRegions package for Julia provides a software framework for the front end of computations over regions in the extended complex plane. It defines abstract types for curves, paths (i.e., piecewise smooth contours), and regions, defining minimal interfaces for them and providing default behavior based only on the interfaces. These are then implemented as concrete subtypes; e.g., a Circle is a subtype of AbstractClosedCurve, and a Polygon is a subtype of AbstractClosedPath. These types define data structures and methods to operate on them.Julia's multiple dispatch facility enables some convenient uses of this basic framework. For example, the Base.intersect method is extended to have definitions for many possible pairings of curve and path arguments. In the future, a method for constructing conformal maps, say, could have one method for arguments of types AbstractDisk and PolygonalRe gion that would call upon a Schwarz-Christoffel mapping code. The end user need not know what sort of mapping algorithm is needed for a particular problem, and the master method could easily be extended to numerous other contexts without the need for a "switchyard" portal that could become difficult to maintain.The ComplexRegions package builds on the ComplexValues package that defines Polar and Spherical types for working with polar and Riemann sphere representations of complex numbers. It also provides recipes for plotting the major abstract types with the popular Plots.jl package. For example:Driscoll, (2019). ComplexRegions.jl: A Julia package for regions in the complex plane.
“…Lipschitz continuous boundaries admit corners, specifically they allow for countable number of corners with the angle θ of each corner satisfying 0 < θ < 2π (thus prohibiting cusps). Solving PDEs on Lipschitz domains to high accuracy everywhere in the domain is in general quite challenging, independent of the numerical method used; see for example the discussion in [7]. When solving boundary integral equations on domains with corners, the standard Nyström method fails to achieve optimal accuracy [3].…”
Two-dimensional Stokes flow through a periodic channel is considered. The channel walls need only be Lipschitz continuous, in other words they are allowed to have corners. Boundary integral methods are an attractive tool for numerically solving the Stokes equations, as the partial differential equation can be reformulated into an integral equation that must be solved only over the boundary of the domain. When the boundary is at least C 1 smooth, the boundary integral kernel is a compact operator, and traditional Nyström methods can be used to obtain highly accurate solutions. In the case of Lipschitz continuous boundaries, however, obtaining accurate solutions using the standard Nyström method can require high resolution. We adapt a technique known as recursively compressed inverse preconditioning to accurately solve the Stokes equations without requiring any more resolution than is needed to resolve the boundary. Combined with a periodic fast summation method we construct a method that is O(N log N) where N is the number of quadrature points on the boundary. We demonstrate the robustness of this method by extending an existing boundary integral method for viscous drops to handle the movement of drops near corners.
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