The field of numerical methods for partial differential equations is witnessing significant developments, driven by the need for more accurate and efficient solutions to complex problems. Researchers are exploring innovative techniques to improve the accuracy and robustness of numerical methods, such as the use of machine learning algorithms, new finite element methods, and advanced discretization techniques. One notable trend is the increasing use of data-driven approaches to solve partial differential equations, which has led to the development of novel methods that can achieve high accuracy on coarse meshes. Another area of focus is the development of more efficient and stable numerical methods for solving time-dependent problems, including the use of tensor neural networks and subordination-based approximations. Noteworthy papers in this area include the work on entropy stable shock capturing for high-order DGSEM on moving meshes, which demonstrates the potential for improved accuracy and stability in simulations of complex flows. The paper on learning high-accuracy numerical schemes for hyperbolic equations on coarse meshes is also significant, as it shows how data-driven approaches can be used to develop more efficient and accurate numerical methods.
Advancements in Numerical Methods for Partial Differential Equations
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A posteriori error estimates for the finite element discretization of second-order PDEs set in unbounded domains
A numerical Bernstein splines approach for nonlinear initial value problems with Hilfer fractional derivative
Improvement of conformal maps combined with the Sinc approximation for derivatives over infinite intervals
Spectral coefficient learning physics informed neural network for time-dependent fractional parametric differential problems
Entropy stable shock capturing for high-order DGSEM on moving meshes
Stable fully discrete finite element methods with BGN tangential motion for Willmore flow of planar curves
Asymptotically accurate and locking-free finite element implementation of the refined shell theory
Convergence of Calder\'on residuals
A simple and general framework for the construction of exactly div-curl-grad compatible discontinuous Galerkin finite element schemes on unstructured simplex meshes
Learning high-accuracy numerical schemes for hyperbolic equations on coarse meshes
A posteriori error analysis of a robust virtual element method for stress-assisted diffusion problems
Subordination based approximation of Caputo fractional propagator and related numerical methods
A Fast Fourth-Order Cut Cell Method for Solving Elliptic Equations in Two-Dimensional Irregular Domains
A Conic Transformation Approach for Solving the Perspective-Three-Point Problem
Solving Time-Fractional Partial Integro-Differential Equations Using Tensor Neural Networks
Error analysis of the diffuse domain finite element method for second order parabolic equations