Piecewise constant collocation for first-kind boundary integral equations

Extract

We wish to correct a minor error in the recent paper [2]. That paper was concerned with an integral equation defined on a closed polygon Γ with r corners at the points x₀, x₂, …, x_2r = x₀. We parameterized Γ using a mapping γ:[−π,π] → Γ defined as follows. For each l, introduce the mid-point x_2l−1 of the side joining x_2l—2 to x_2l. Then introduce 2r + 1 points in parameter space

with the property that for each j = 1, …, 2r

where m_j are integers and . Then γ(s) is defined by

for j = 1, …, 2r. The {S_j} are then the preimages of the {x_j} under γ. Moreover, in view of (1), a family of uniform meshes can be constructed on [−π, π] which include {S_j} as the break-points. Then γ maps these to meshes which are uniform on each segment joining x_j−1 to x_j (which we denote Γ_j). These meshes are used to discretize the integral equation.