The field of numerical methods for partial differential equations is rapidly advancing, with a focus on developing innovative and efficient methods for solving complex problems. Recent developments have centered around improving the accuracy and stability of existing methods, such as discontinuous Galerkin methods and finite element methods, as well as exploring new approaches like virtual element methods and meshfree methods. Notable advancements include the development of high-order methods, anisotropic mesh adaptation, and structure-preserving methods. These advancements have significant implications for a wide range of applications, from fluid dynamics and solid mechanics to inverse problems and optimization. Some noteworthy papers in this area include the development of efficient optimization-based invariant-domain-preserving limiters and the proposal of a multiscale spectral generalized finite element method for discontinuous Galerkin schemes. Additionally, the introduction of a grad-curl conforming virtual element method and a fast spectral overlapping domain decomposition method have shown promising results. Overall, the field is moving towards the development of more accurate, efficient, and robust numerical methods for solving complex partial differential equations.
Advancements in Numerical Methods for Partial Differential Equations
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Efficient optimization-based invariant-domain-preserving limiters in solving gas dynamics equations
Multiscale Spectral Generalized Finite Element Methods for Discontinuous Galerkin Schemes
A numerical method for the fractional Zakharov-Kuznetsov equation
Anisotropic mesh adaptation for unsteady two-phase flow simulation with the Cahn-Hilliard Navier-Stokes model
Nonconforming Linear Element Method for a Generalized Tensor-Valued Stokes Equation with Application to the Triharmonic Equation
HoSGFEM: High-order stable generalized finite element method for elliptic interface problem
Effective numerical integration on complex shaped elements by discrete signed measures
Multiscale modeling for contact problem with high-contrast heterogeneous coefficients with primary-dual formulation
Stability analysis of discontinuous Galerkin with a high order embedded boundary treatment for linear hyperbolic equations
An Energy-Stable Discontinuous Galerkin Method for the Compressible Navier--Stokes--Allen--Cahn System
A least squares finite element method for backward parabolic problems
A grad-curl conforming virtual element method for a grad-curl problem linking the 3D quad-curl problem and Stokes system
A virtual element approximation for the modified transmission eigenvalues for natural materials
A structure-preserving Lagrangian discontinuous Galerkin method using flux and slope limiting
A fast spectral overlapping domain decomposition method with discretization-independent conditioning bounds
A two-dimensional fractional-order element-free Galerkin method for nonlocal elasticity and complex domain problems
Asymptotic meshes from $r$-variational adaptation methods for static problems in one dimension
A GenEO-type coarse space with smaller eigenproblems