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Geometrically Graded h-p Quadrature Applied to the Complex Boundary Integral Equation Method for the Dirichlet Problem with Corner Singularities

David De Wit

Abstract

Boundary integral methods for the solution of boundary value PDEs are an alternative to `interior' methods, such as finite difference and finite element methods. They are attractive on domains with corners, particularly when the solution has singularities at these corners. In these cases, interior methods can become excessively expensive, as they require a finely discretised 2D mesh in the vicinity of corners, whilst boundary integral methods typically require a mesh discretised in only one dimension, that of arc length. Consider the Dirichlet problem. Traditional boundary integral methods applied to problems with corner singularities involve a (real) boundary integral equation with a kernel containing a logarithmic singularity. This is both tedious to code and computationally inefficient. The CBIEM is different in that it involves a complex boundary integral equation with a smooth kernel. The boundary integral equation is approximated using a collocation technique, and the interior solution is then approximated using a discretisation of Cauchy's integral formula, combined with singularity subtraction. A high order quadrature rule is required for the solution of the integral equation. Typical corner singularities are of square root type, and a `geometrically graded h-p' composite quadrature rule is used. This yields efficient, high order solution of the integral equation, and thence the Dirichlet problem. Implementation and experimental results in \textsc{matlab} code are presented.

Geometrically Graded h-p Quadrature Applied to the Complex Boundary Integral Equation Method for the Dirichlet Problem with Corner Singularities

Abstract

Boundary integral methods for the solution of boundary value PDEs are an alternative to `interior' methods, such as finite difference and finite element methods. They are attractive on domains with corners, particularly when the solution has singularities at these corners. In these cases, interior methods can become excessively expensive, as they require a finely discretised 2D mesh in the vicinity of corners, whilst boundary integral methods typically require a mesh discretised in only one dimension, that of arc length. Consider the Dirichlet problem. Traditional boundary integral methods applied to problems with corner singularities involve a (real) boundary integral equation with a kernel containing a logarithmic singularity. This is both tedious to code and computationally inefficient. The CBIEM is different in that it involves a complex boundary integral equation with a smooth kernel. The boundary integral equation is approximated using a collocation technique, and the interior solution is then approximated using a discretisation of Cauchy's integral formula, combined with singularity subtraction. A high order quadrature rule is required for the solution of the integral equation. Typical corner singularities are of square root type, and a `geometrically graded h-p' composite quadrature rule is used. This yields efficient, high order solution of the integral equation, and thence the Dirichlet problem. Implementation and experimental results in \textsc{matlab} code are presented.

Paper Structure

This paper contains 53 sections, 4 theorems, 80 equations, 9 figures, 10 tables.

Table of Contents

  1. Introduction
  2. $h$-$p$ Quadrature Methods
  3. Introduction
  4. Quadrature Methods -- the Questions
  5. Gaußian Quadrature
  6. Graded Meshes
  7. $h$, $p$ and $h$-$p$ Quadrature Methods
  8. $h$ Methods
  9. $p$ Methods
  10. $h$-$p$ Methods
  11. Error Analysis for the $h$ and $h$-$p$ Methods
  12. Error Analysis for the $h$ Method
  13. Error Analysis for the $h$-$p$ Method
  14. Experimental Results for a Real Integral
  15. Extension to Complex Contour Integrals
  16. ...and 38 more sections

Key Result

Theorem 2.1

Given $f \in C^{2n-2} \left[ a, b \right]$, the $n$ point Gauß--Lobatto quadrature rule ($n \geqslant 2$), has nodes $a \equiv x_1 < x_2 < \dots < x_{n-1} < x_n \equiv b$, and positive weights $w_1, \dots, w_n$ such that: Here $E_n$ is dependent on $f$, $a$, $b$ and $n$: The rule is of degree $p = 2n - 3$ (this is always odd). Observe that for $n = 2$ the rule is the trapezoidal rule, and for $n

Figures (9)

  • Figure 1: Errors for $h$ and $h$-$p$ methods applied to $\int_0^1 \sqrt{x} dx$, for various choices of $\gamma \equiv g$. The slopes of the lines are approximately $- 3 \gamma / 2$. As the error tends to machine precision ($\epsilon \approx 10^{-16}$), the convergence results lose their regularity.
  • Figure 2: Errors for $h$ and $h$-$p$ methods applied to $\int_0^1 \sqrt{x} dx$, for constant $p$. The slopes of the lines are approximately $- \left( p + 1 \right)$.
  • Figure 3: Error results for the complex contour integral.
  • Figure 4: CBIEM nomenclature. Corner, mesh, node and collocation points on contour $\Gamma$, about a region $\Omega$. The lengths marked on segment $1$ are the positions of geometric mesh points, in terms of a unit arc length on that segment. The node points correspond to a closed Newton--Cotes rule on each mesh interval.
  • Figure 5: Structure of $B$.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Theorem 2.1: Gauß--Lobatto Quadrature
  • Theorem 2.2: Convergence of the $h$ Method with Algebraic Grading
  • Theorem 2.1
  • Theorem 2.3: Convergence of the $h$-$p$ Method with Geometric Grading