arXiv subjects

Svetlana Tlupova

Publications and source records attributed to Svetlana Tlupova.

10 recordsLinked to original sources

arXiv2026

High-order kernel regularization of singular and hypersingular Helmholtz boundary integral operators

This paper extends and analyzes the high-order kernel regularization framework of Beale & Tlupova (arXiv:2510.13639) to all four on-surface boundary integral operators of the Helmholtz Calderon calculus in three dimensions: the single-layer, double-layer, adjoint double-layer, and hypersingular operators. To the best of our knowledge, this work provides the first high-order kernel regularization of the hypersingular operator for both the Helmholtz and Laplace equations in three dimensions. The regularization replaces each singular kernel with a smooth modification constructed from error functions together with a polynomial correction whose coefficients are determined through moment conditions. Alongside the derivation of the regularizing functions, the paper provides a unified error analysis of the combined regularization and quadrature discretization procedure. By coupling the regularization parameter to the mesh size, the two error contributions can be balanced, leading to explicit overall convergence rates that depend jointly on the order of the regularization and the degree of exactness of the surface quadrature rule. A key practical feature of the method is its implementation simplicity. Once the regularizing functions are determined, the numerical task reduces entirely to the evaluation of smooth surface integrals using standard quadrature, without the need for element-local solves, singularity-specific precomputations, or specialized quadrature rules. Although the modified kernel is generally incompatible with kernel-specific fast methods, this limitation is addressed through H-matrix acceleration, applicable in a black-box manner. Numerical examples -- including verification of the predicted convergence rates and solution of sound-soft and sound-hard scattering problems by smooth obstacles -- demonstrate the accuracy and practicality of the proposed methodology.

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arXiv2025

High order regularization of nearly singular surface integrals

Solutions of partial differential equations can often be written as surface integrals having a kernel related to a singular fundamental solution. Special methods are needed to evaluate the integral accurately at points on or near the surface. Here we derive formulas to regularize the integrals with high accuracy, using analysis from Beale and Tlupova (Adv. Comput. Math., 2024), so that a standard quadrature can be used without special care near the singularity. We treat single or double layer integrals for harmonic functions or for Stokes flow. The nearly singular case, evaluation at points close to the surface, can be needed when surfaces are close to each other, or to find values at grid points near a surface. We derive formulas for regularized kernels with error $O(δ^p)$ where $δ$ is the smoothing radius and $p = 3$, $5$, $7$. With spacing $h$ in the quadrature, we choose $δ= κh^q$ with $q<1$ so that the discretization error is controlled as $h \to 0$. We see the predicted order of convergence $O(h^{pq})$ in various examples. Values at all grid points can be obtained from those near the surface in an efficient manner suggested in A. Mayo (SIAM J. Statist. Comput., 1985). With this technique we obtain high order accurate grid values for a harmonic function determined by interfacial conditions and for the pressure and velocity in Stokes flow around a translating spheroid.

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arXiv2025

Localized evaluation and fast summation in the extrapolated regularization method for integrals in Stokes flow

Boundary integral equation methods are widely used in the solution of many partial differential equations. The kernels that appear in these surface integrals are nearly singular when evaluated near the boundary, and straightforward numerical integration produces inaccurate results. In Beale and Tlupova (Adv. Comput. Math, 2024), an extrapolated regularization method was proposed to accurately evaluate the nearly singular single and double-layer surface integrals for harmonic potentials or Stokes flow. The kernels are regularized using a smoothing parameter, and then a standard quadrature is applied. The integrals are computed for three choices of the smoothing parameter to find the extrapolated value to fifth order accuracy. In this work, we apply several techniques to reduce the computational cost of the extrapolated regularization method applied to the Stokes single and double layer integrals. First, we use a straightforward OpenMP parallelization over the target points. Second, we note that the effect of the regularization is local and evaluate only the local component of the sum for three values of the smoothing parameter. The non-local component of the sum is only evaluated once and reused in the other sums. This component is still the computational bottleneck as it is $O(N^2)$, where $N$ is the system size. We apply the kernel-independent treecode to these far-field interactions to reduce the CPU time. We carry out experiments to determine optimal parameters both in terms of accuracy and efficiency of the computations. We then use these techniques to compute Stokes flow around two spheres that are nearly touching.

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arXiv2024

Extrapolated regularization of nearly singular integrals on surfaces

We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral equation when one surface is close to another or to obtain values at grid points. We replace the singular kernel with a regularized version having a length parameter $δ$ in order to control discretization error. Analysis near the singularity leads to an expression for the error due to regularization which has terms with unknown coefficients multiplying known quantities. By computing the integral with three choices of $δ$ we can solve for an extrapolated value that has regularization error reduced to $O(δ^5)$, uniformly for target points on or near the surface. In examples with $δ/h$ constant and moderate resolution we observe total error about $O(h^5)$ close to the surface. For convergence as $h \to 0$ we can choose $δ$ proportional to $h^q$ with $q < 1$ to ensure the discretization error is dominated by the regularization error. With $q = 4/5$ we find errors about $O(h^4)$. For harmonic potentials we extend the approach to a version with $O(δ^7)$ regularization; it typically has smaller errors but the order of accuracy is less predictable.

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arXiv2023

The adjoint double layer potential on smooth surfaces in $\mathbb{R}^3$ and the Neumann problem

We present a simple yet accurate method to compute the adjoint double layer potential, which is used to solve the Neumann boundary value problem for Laplace's equation in three dimensions. An expansion in curvilinear coordinates leads us to modify the expression for the adjoint double layer so that the singularity is reduced when evaluating the integral on the surface. We then regularize the Green's function, with a radial parameter $δ$. We show that a natural regularization has error $O(δ^3)$, and a simple modification improves the error to $O(δ^5)$. The integral is evaluated numerically without the need of special coordinates. We use this treatment of the adjoint double layer to solve the classical integral equation for the interior Neumann problem and evaluate the solution on the boundary. Choosing $δ= ch^{4/5}$, we find about $O(h^4)$ convergence in our examples, where $h$ is the spacing in a background grid.

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arXiv2021

A Novel Regularization for Higher Accuracy in the Solution of 3D Stokes Flow

Many problems in fluid dynamics are effectively modeled as Stokes flows - slow, viscous flows where the Reynolds number is small. Boundary integral equations are often used to solve these problems, where the fundamental solutions for the fluid velocity are the Stokeslet and stresslet. One of the main challenges in evaluating the boundary integrals is that the kernels become singular on the surface. A regularization method that eliminates the singularities and reduces the numerical error through correction terms for both the Stokeslet and stresslet integrals was developed in Tlupova and Beale, JCP (2019). In this work we build on the previously developed method to introduce a new stresslet regularization that is simpler and results in higher accuracy when evaluated on the surface. Our regularization replaces a seventh-degree polynomial that results from an equation with two conditions and two unknowns with a fifth-degree polynomial that results from an equation with one condition and one unknown. Numerical experiments demonstrate that the new regularization retains the same order of convergence as the regularization developed by Tlupova and Beale but shows a decreased magnitude of the error.

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arXiv2021

A domain decomposition solution of the Stokes-Darcy system in 3D based on boundary integrals

A framework is developed for a robust and highly accurate numerical solution of the coupled Stokes-Darcy system in three dimensions. The domain decomposition method is based on a Dirichlet-Neumann type splitting of the interface conditions and solving separate Stokes and Darcy problems iteratively. Second kind boundary integral equations are formulated for each problem. The integral equations use a smoothing of the kernels that achieves high accuracy on the boundary, and a straightforward quadrature to discretize the integrals. Numerical results demonstrate the convergence, accuracy, and dependence on parameter values of the iterative solution for a problem of viscous flow around a porous sphere with a known analytical solution, as well as more general surfaces.

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arXiv2019

A kernel-independent treecode based on barycentric Lagrange interpolation

A kernel-independent treecode (KITC) is presented for fast summation of particle interactions. The method employs barycentric Lagrange interpolation at Chebyshev points to approximate well-separated particle-cluster interactions. The KITC requires only kernel evaluations, is suitable for non-oscillatory kernels, and it utilizes a scale-invariance property of barycentric Lagrange interpolation. For a given level of accuracy, the treecode reduces the operation count for pairwise interactions from $O(N^2)$ to $O(N\log N)$, where $N$ is the number of particles in the system. The algorithm is demonstrated for systems of regularized Stokeslets and rotlets in 3D, and numerical results show the treecode performance in terms of error, CPU time, and memory consumption. The KITC is a relatively simple algorithm with low memory consumption, and this enables a straightforward OpenMP parallelization.

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arXiv2019

A Treecode Algorithm for 3D Stokeslets and Stresslets

The Stokeslet and stresslet kernels are commonly used in boundary element simulations and singularity methods for slow viscous flow. Evaluating the velocity induced by a collection of Stokeslets and stresslets by direct summation requires $O(N^2)$ operations, where $N$ is the system size. The present work develops a treecode algorithm for 3D Stokeslets and stresslets that reduces the cost to $O(N\log N)$. The particles are divided into a hierarchy of clusters, and well-separated particle-cluster interactions are computed by a far-field Cartesian Taylor approximation. The terms in the approximation are contracted to promote efficient computation. Serial and parallel results display the treecode performance for several test cases. In particular the method has relatively simple structure and low memory usage, and this enhances parallel efficiency for large systems.

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arXiv2019

Regularized Single and Double Layer Integrals in 3D Stokes Flow

We present a numerical method for computing the single layer (Stokeslet) and double layer (stresslet) integrals in Stokes flow. The method applies to smooth, closed surfaces in three dimensions, and achieves high accuracy both on and near the surface. The singular Stokeslet and stresslet kernels are regularized and, for the nearly singular case, corrections are added to reduce the regularization error. These corrections are derived analytically for both the Stokeslet and the stresslet using local asymptotic analysis. For the case of evaluating the integrals on the surface, as needed when solving integral equations, we design high order regularizations for both kernels that do not require corrections. This approach is direct in that it does not require grid refinement or special quadrature near the singularity, and therefore does not increase the computational complexity of the overall algorithm. Numerical tests demonstrate the uniform convergence rates for several surfaces in both the singular and near singular cases, as well as the importance of corrections when two surfaces are close to each other.

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