The field of numerical methods for partial differential equations (PDEs) is rapidly advancing, with a focus on developing innovative and efficient methods for solving complex problems. Recent developments have centered around improving the stability and accuracy of existing methods, such as the discontinuous Galerkin method and finite element methods. Additionally, there is a growing interest in using machine learning techniques, such as neural operators, to solve PDEs and improve the interpretability of results. Noteworthy papers include the introduction of RED-DiffEq, a framework that leverages pretrained diffusion models for solving inverse PDE problems, and the development of GeoFunFlow, a geometric diffusion model framework for inverse problems on complex geometries. These advancements have the potential to significantly impact various fields, including physics, engineering, and computer science.
Advances in Numerical Methods for PDEs
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Towards provable energy-stable overset grid methods using sub-cell summation-by-parts operators
Conforming lifting and adjoint consistency for the Discrete de Rham complex of differential forms
RED-DiffEq: Regularization by denoising diffusion models for solving inverse PDE problems with application to full waveform inversion
The discretizations of the derivative by the continuous Galerkin and the discontinuous Galerkin methods are exactly the same
Finite Element Complexes with Traces Structures: A unified framework for cohomology and bounded interpolation
GeoFunFlow: Geometric Function Flow Matching for Inverse Operator Learning over Complex Geometries
Numerical and analytical modeling of heat equation in current-carrying conductors using the heat equation implemented using Finite-JAX
Diffuse Domain Methods with Dirichlet Boundary Conditions
A bound-preserving multinumerics scheme for steady-state convection-diffusion equations
New Fourth-Order Grayscale Indicator-Based Telegraph Diffusion Model for Image Despeckling
Finite element discretizations of bending plates with prestrained microstructure
Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling
A posteriori error estimation for weak Galerkin method of the fourth-order singularly perturbed problem
A multi-resolution limiter for the Runge-Kutta discontinuous Galerkin method
Time-marching multi-level variational multiscale tensor decomposition algorithm for heat conduction with moving heat source
A Computationally Efficient Finite Element Method for Shape Reconstruction of Inverse Conductivity Problems
COMMET: orders-of-magnitude speed-up in finite element method via batch-vectorized neural constitutive updates
Implementation techniques for multigrid solvers for high-order Discontinuous Galerkin methods
A nodally bound-preserving composite discontinuous Galerkin method on polytopic meshes