Numerical Methods in Geometric Function Theory

Numerical methods in geometric function theory focus on the algorithmic computation of conformal mappings and related invariants for planar domains. Central to these approaches are boundary integral equation techniques, series expansions and potential‐theoretic discretisations, which together convert classical boundary value problems in complex analysis into tractable numerical schemes. Boundary integral methods, often accelerated by the fast multipole method or specialised preconditioners, deliver high‐precision results for domains with smooth, piecewise‐smooth or even polygonal boundaries. Series‐based approaches employ Fourier, Laurent or orthogonal polynomial expansions to approximate maps between canonical domains and more general geometries, while charge simulation and Schottky–Klein functions extend these ideas to multiply connected regions. Emphasis on rigorous error estimation, adaptive meshing and software toolboxes has broadened the accessibility of these methods, enabling applications across fluid mechanics, electromagnetism, materials science and computational geometry.

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Research from all publishers

Recent work has advanced numerical conformal mapping of polycircular domains by combining boundary integral equations with the fast multipole method, yielding sharp error estimates and convergence analyses for moduli of quadrilaterals in domains bounded by circular arcs. Novel computational algorithms integrating hyperbolic geometry and fast multipole accelerations have been developed to compute condenser capacities and hyperbolic perimeters, demonstrating exceptionally high precision on model problems and confirming sharpness of theoretical inequalities. A versatile MATLAB toolbox has been introduced for mapping polygonal multiply connected domains onto circular domains, supporting high connectivity and complex obstacle arrangements while offering user‐friendly interfaces and robust inverse‐map computations.

Numerical Methods in Geometric Function Theory publication trend

The graph below shows the total number of articles in numerical methods in geometric function theory across all publications each year (not limited to Nature Index journals).

Technical terms

Conformal mapping: Angle‐preserving bijective transformation between planar domains that is holomorphic and has a nonzero derivative.

Boundary integral equation: Reformulation of a boundary value problem as an integral equation over the domain boundary, reducing dimensionality and concentrating computational effort on the boundary.

Multiply connected domain: A planar region whose complement has more than one connected component, requiring specialised treatment of each boundary component in mapping algorithms.

Conformal invariant: Quantity associated with a domain or mapping (such as capacity, modulus or harmonic measure) that remains unchanged under conformal transformations.

Fast multipole method: Hierarchical algorithm that accelerates the evaluation of long‐range interactions in boundary integral formulations, reducing computational complexity from quadratic to nearly linear.

Neumann kernel: Integral kernel arising in boundary integral equations for Laplace’s equation, used to enforce Neumann (flux) boundary conditions in conformal mapping computations.

References

1. Polycircular domains, numerical conformal mappings, and moduli of quadrilaterals. Advances in Computational Mathematics (2022).
2. Condenser capacity and hyperbolic perimeter Image 1. Computers & Mathematics with Applications (2022).
3. PlgCirMap: A MATLAB toolbox for computing conformal mappings from polygonal multiply connected domains onto circular domains. SoftwareX (2020).