Advances in Numerical Methods for Elliptic Problems

The field of numerical methods for elliptic problems is rapidly evolving, with a focus on developing innovative and efficient methods for solving complex problems. Recent developments have centered around the creation of new numerical methods, such as kernel-free boundary integral methods and immersed penalized boundary methods, which have shown promise in solving elliptic interface problems and boundary value problems on curved domains. Additionally, there has been a growing interest in using machine learning techniques to improve the accuracy and efficiency of numerical methods. Noteworthy papers in this area include: A kernel-free boundary integral method for elliptic interface problems on surfaces, which presents a generalized boundary integral method for elliptic equations on surfaces. Walk-on-Interfaces: A Monte Carlo Estimator for an Elliptic Interface Problem, which introduces a grid-free Monte Carlo estimator for a class of Neumann elliptic interface problems with nonhomogeneous flux jump conditions. Energy minimisation using overlapping tensor-product free-knot B-splines, which studies a nonlinear approximation scheme based on overlapping tensor-product free-knot B-spline patches for solving PDEs with localized features.

Sources

A kernel-free boundary integral method for elliptic interface problems on surfaces

Using the Immersed Penalized Boundary Method with Splines to Solve PDE's on Curved Domains in 3D

Walk-on-Interfaces: A Monte Carlo Estimator for an Elliptic Interface Problem with Nonhomogeneous Flux Jump Conditions and a Neumann Boundary Condition

Harmonic potentials in the de Rham complex

Some new properties of the PamPa scheme

Energy minimisation using overlapping tensor-product free-knot B-splines

A parametric finite element method for the incompressible Navier--Stokes equations on an evolving surface

A new class of regularized preconditioners for double saddle-point problems

Eighth-Order Accurate Methods for Boundary Value Problems Arising from the Lane-Emden Equation

A comparative study of finite element methods for a class of harmonic map heat flow problems

Curvilinear coordinates and curvature in radiative transport

Uniform error analysis of a rectangular Morley finite element method on a Shishkin mesh for a 4th-order singularly perturbed boundary value problem

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