Advances in Numerical Methods for Partial Differential Equations

The field of numerical methods for partial differential equations is rapidly advancing, with a focus on developing more efficient and accurate methods for solving complex problems. One of the key areas of research is the development of new preconditioning techniques for saddle-point systems, which are commonly encountered in fluid dynamics and other fields. Researchers are also exploring new methods for solving eigenvalue problems, including the use of inexact iteration methods and locally optimal block preconditioned conjugate gradient methods. Additionally, there is a growing interest in the development of new methods for solving problems on non-smooth domains, including the use of polar parameterizations and graded mesh refinement. Notable papers in this area include the development of optimal transfer operators for algebraic two-level methods and the creation of a framework for analyzing discrete exterior calculus approximations to Hodge-Laplacian problems. Particularly noteworthy papers include:

  • Optimal transfer operators in algebraic two-level methods for nonsymmetric and indefinite problems, which significantly strengthens previous results and provides a new theory for optimal transfer operators.
  • A Framework for Analysis of DEC Approximations to Hodge-Laplacian Problems using Generalized Whitney Forms, which provides a new framework for interpreting discrete exterior calculus numerical schemes and rigorously proves convergence with rates for the Hodge-Laplacian problem.

Sources

Optimal transfer operators in algebraic two-level methods for nonsymmetric and indefinite problems

Stable fully practical finite element methods for axisymmetric Willmore flow

Triangular preconditioners for double saddle point linear systems arising in the mixed form of poroelasticity equations

A Convergent Inexact Abedin-Kitagawa Iteration Method for Monge-Amp\`ere Eigenvalue Problems

Optimal pressure approximation for the nonstationary Stokes problem by a variational method in time with post-processing

Field of values analysis that includes zero for preconditioned nonsymmetric saddle-point systems

Local Convergence Behavior of Extended LOBPCG for Computing Eigenvalues of Hermitian Matrices

A Reynolds-semi-robust H(div)-conforming method for unsteady incompressible non-Newtonian flows

A Framework for Analysis of DEC Approximations to Hodge-Laplacian Problems using Generalized Whitney Forms

Convergence and Stability of Discrete Exterior Calculus for the Hodge Laplace Problem in Two Dimensions

Vertex-based auxiliary space multigrid method and its application to linear elasticity equations

Error Estimates and Graded Mesh Refinement for Isogeometric Analysis on Polar Domains with Corners

A general regularization strategy for singular Stokes problems and convergence analysis for corresponding discretization and iterative solution

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