The field of numerical methods for partial differential equations is witnessing significant advancements, driven by the need for more accurate, efficient, and robust solutions to complex problems. A key direction in this area is the development of innovative discretization techniques, such as the application of weak Galerkin methods to non-convex polytopal meshes and the use of smoothed finite element methods for electro-mechanically coupled problems. Another important trend is the improvement of existing methods, including the enhancement of finite difference methods for nonlinear convection-diffusion equations and the development of more efficient preconditioners for high-order schemes arising from multi-dimensional Riesz space fractional diffusion equations. Furthermore, researchers are exploring new approaches to solve specific types of equations, such as the Helmholtz equation, using algorithms like the Multi-Frequency WaveHoltz method. Noteworthy papers in this regard include the proposal of a non-iterative domain decomposition time integrator for linear wave equations and the development of an optimal preconditioner for high-order schemes arising from multi-dimensional Riesz space fractional diffusion equations with variable coefficients. Overall, these advancements are expected to have a significant impact on various fields, including physics, engineering, and biomedical research, by enabling more accurate simulations and predictions.
Advancements in Numerical Methods for Partial Differential Equations
Sources
A conservative invariant-domain preserving projection technique for hyperbolic systems under adaptive mesh refinement
Fourth-Order Compact FDMs for Steady and Time-Dependent Nonlinear Convection-Diffusion Equations
A Simple and Robust Weak Galerkin Method for the Brinkman Equations on Non-Convex Polytopal Meshes
A non-iterative domain decomposition time integrator for linear wave equations
Convergence of Discrete Exterior Calculus for the Hodge-Dirac Operator
Implementation and Basis Construction for Smooth Finite Element Spaces
Time-continuous strongly conservative space-time finite element methods for the dynamic Biot model
A fixed-time stable dynamical model for solving EVLCPs
Divergence-free Preserving Mix Finite Element Methods for Fourth-order Active Fluid Model
An inherent regularization approach to parameter-free preconditioning for nearly incompressible linear poroelasticity and elasticity
Pulsatile Flows for Simplified Smart Fluids with Variable Power-Law: Analysis and Numerics
Space-time finite element methods for nonlinear wave equations via elliptic regularisation
Modelling and simulation of electro-mechanically coupled dielectric elastomers and myocardial tissue using smoothed finite element methods
Numerical Fredholm determinants for matrix-valued kernels on the real line
$hp$-adaptive finite element simulation of a static anti-plane shear crack in a nonlinear strain-limiting elastic solid
An optimal preconditioner for high-order scheme arising from multi-dimensional Riesz space fractional diffusion equations with variable coefficients
Fitted norm preconditioners for the Hodge Laplacian in mixed form
A Multi-Frequency Helmholtz Solver Based on the WaveHoltz Algorithm