arXiv subjects

Gentian Zavalani

Publications and source records attributed to Gentian Zavalani.

9 recordsLinked to original sources

arXiv2026

Odd behaviour of even geometries: an explanation for superconvergent geometric consistency errors

Piecewise polynomial surface approximations used in surface finite element methods often seem to behave better than their standard approximation properties suggest if their polynomial order is even. We explain this superconvergence through cancellation of leading interpolation errors on suitably structured meshes that naturally arise in some refinement processes. This cancellation improves weighted integral estimates for functions, derivatives, and geometric quantities. Applications include estimates for surface normals, the Weingarten map, and Gaussian curvature. Numerical experiments reproduce the predicted parity-dependent behaviour and support the proposed explanation of superconvergent geometric consistency errors, while the corresponding pointwise errors retain their standard orders.

math.NA↗

arXiv2026

Cubature from rational approximation

We present a numerical construction of cubature rules for area integrals of analytic functions over planar domains with rectifiable Jordan boundary. The starting point is the Cauchy--Green identity. Given a weight $w$, we choose a $\bar\partial$-antiderivative $W$ and reduce the area integral to a contour integral involving the boundary values of $W$. These values are then approximated by a rational function with free poles, computed by the AAA algorithm. The poles inside the domain become cubature nodes, the corresponding residues become weights, and the boundary residual controls the error through an a posteriori estimate, rigorous once the continuous boundary residual is bounded. The same rule admits a dual reading, as the exact integral of a rational interpolant to the integrand, the area analogue of the one-dimensional interpolatory viewpoint. The numerical examples recover the disk mean-value rule and the focal-segment rule of the ellipse to machine precision, reproduce the exact finite quadrature identities of quadrature domains with both separated and confluent nodes, and evaluate logarithmic and Cauchy volume potentials from boundary data alone. The interior poles trace analytic skeletons that we identify tentatively with the mother bodies of potential theory, along with image points that appear without being imposed; for the square the observed convergence is root-exponential.

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arXiv2026

The Bojanov--Naidenov inequality for quartics and second derivatives

We settle the case $n=4$, $k=2$ of the Bojanov--Naidenov problem for algebraic polynomials. Let $P$ be a real polynomial of degree at most four with $\left\lVert{P}\right\rVert_{C[-1,1]}\leq 1$, and let $T_4(x)=8x^4-8x^2+1$. We prove that, for every $t\geq0$, \[ \int_{-1}^{1} \bigl(|P''(x)|-t\bigr)_+\,dx \leq \int_{-1}^{1} \bigl(|T_4''(x)|-t\bigr)_+\,dx . \] This tail estimate implies \[ \int_{-1}^{1}φ(|P''(x)|)\,dx \leq \int_{-1}^{1}φ(|T_4''(x)|)\,dx \] for every nondecreasing convex function $φ:[0,\infty)\to\mathbb{R}$. If $φ$ is strictly increasing and convex, equality can occur only for $P=\pm T_4$. The proof is elementary and finite. We interpolate at the five extremal points of $T_4$; convexity then reduces the problem to the $32$ sign choices at these nodes. At each vertex the second derivative is a quadratic polynomial, so the remaining work is an explicit comparison of level sets.

math.NA↗

arXiv2026

A High-Order Fast Direct Solver for Surface PDEs on Triangles

We develop a triangular formulation of the hierarchical Poincaré-Steklov (HPS) method for elliptic partial differential equations on surfaces, allowing high-order discretizations on unstructured meshes and complex geometries. Classical HPS formulations rely on high-order quadrilateral meshes and tensor-product spectral discretizations, which enable efficient algorithms but restrict applicability to structured geometries. To overcome this restriction, we introduce a triangle-based hierarchical Poincaré-Steklov scheme (THPS) built on orthogonal Dubiner polynomial bases. As in the classical HPS framework, local solution operators and Dirichlet-to-Neumann maps are constructed and merged hierarchically, yielding a fast direct solver with $O(N \log N)$ complexity for repeated solves on meshes with $N$ elements. The reuse of precomputed operators makes the method particularly effective for implicit time-stepping of surface PDEs. Numerical experiments demonstrate that the proposed method retains spectral accuracy and achieves high-order convergence for a range of static and time-dependent test problems.

math.NA↗

arXiv2025

Fast high-order spectral solvers for PDEs on triangulated surfaces with applications to deforming surfaces

In this paper, we extend the classical quadrilateral based hierarchical Poincaré-Steklov (HPS) framework to triangulated geometries. Traditionally, the HPS method takes as input an unstructured, high-order quadrilateral mesh and relies on tensor-product spectral discretizations on each element. To overcome this restriction, we introduce two complementary high-order strategies for triangular elements: a reduced quadrilateralization approach which is straightforward to implement, and triangle based spectral element method based on Dubiner polynomials. We show numerically that these extensions preserve the spectral accuracy, efficiency, and fast direct-solver structure of the HPS framework. The method is further extended to time dependent and evolving surfaces, and its performance is demonstrated through numerical experiments on reaction-diffusion systems, and geometry driven surface evolution.

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arXiv2025

High-order integration on regular triangulated manifolds reaches super-algebraic approximation rates through cubical re-parameterizations

We present a novel methodology for deriving high-order volume elements (HOVE) designed for the integration of scalar functions over regular embedded manifolds. For constructing HOVE we introduce square-squeezing --a homeomorphic multilinear hypercube-simplex transformation reparametrizing an initial flat triangulation of the manifold to a cubical mesh. By employing square-squeezing, we approximate the integrand and the volume element for each hypercube domain of the reparameterized mesh through interpolation in Chebyshev-Lobatto grids. This strategy circumvents the Runge phenomenon, replacing the initial integral with a closed-form expression that can be precisely computed by high-order quadratures. We prove novel bounds of the integration error in terms of the $r^\text{th}$-order total variation of the integrand and the surface parameterization, predicting high algebraic approximation rates that scale solely with the interpolation degree and not, as is common, with the average simplex size. For smooth integrals whose total variation is constantly bounded with increasing $r$, the estimates prove the integration error to decrease even exponentially, while mesh refinements are limited to achieve algebraic rates. The resulting approximation power is demonstrated in several numerical experiments, particularly showcasing $p$-refinements to overcome the limitations of $h$-refinements for highly varying smooth integrals.

math.NA↗

arXiv2024

High-order numerical integration on regular embedded surfaces

We present a high-order surface quadrature (HOSQ) for accurately approximating regular surface integrals on closed surfaces. The initial step of our approach rests on exploiting square-squeezing--a homeomorphic bilinear square-simplex transformation, re-parametrizing any surface triangulation to a quadrilateral mesh. For each resulting quadrilateral domain we interpolate the geometry by tensor polynomials in Chebyshev--Lobatto grids. Posterior the tensor-product Clenshaw-Curtis quadrature is applied to compute the resulting integral. We demonstrate efficiency, fast runtime performance, high-order accuracy, and robustness for complex geometries.

math.NA↗

arXiv2024

A note on the rate of convergence of integration schemes for closed surfaces

In this paper, we issue an error analysis for integration over discrete surfaces using the surface parametrization presented in [PS22] as well as prove why even-degree polynomials exhibit a higher convergence rate than odd-degree polynomials. Additionally, we provide some numerical examples that illustrate our findings and propose a potential approach that overcomes the problems associated with the original one.

math.NA↗

arXiv2022

Global Polynomial Level Sets for Numerical Differential Geometry of Smooth Closed Surfaces

We present a computational scheme that derives a global polynomial level set parametrisation for smooth closed surfaces from a regular surface-point set and prove its uniqueness. This enables us to approximate a broad class of smooth surfaces by affine algebraic varieties. From such a global polynomial level set parametrisation, differential-geometric quantities like mean and Gauss curvature can be efficiently and accurately computed. Even 4$^{\text{th}}$-order terms such as the Laplacian of mean curvature are approximates with high precision. The accuracy performance results in a gain of computational efficiency, significantly reducing the number of surface points required compared to classic alternatives that rely on surface meshes or embedding grids. We mathematically derive and empirically demonstrate the strengths and the limitations of the present approach, suggesting it to be applicable to a large number of computational tasks in numerical differential geometry.

math.NA↗