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1. Comparison of variational discretizations for a convection-diffusion problem

   Bacuta, Constantin; Bacuta, Cristina; Hayes, Daniel (December 2024, Revue Roumaine Mathématiques Pures et Appliquées)

   Beznea, Lucian; Putinar, Mihai (Ed.)

   For a model convection-diffusion problem, we obtain new error estimates for a general upwinding finite element discretization based on bubble modification of the test space. The key analysis tool is finding representations of the optimal norms on the trial spaces at the continuous and discrete levels. We analyze and compare three methods: the standard linear discretization, the saddle point least square and the upwinding Petrov-Galerkin methods. We conclude that the bubble upwinding Petrov-Galerkin method is the most performant discretization for the one dimensional model. Our results for the model convection-diffusion problem can be extended for creating new and efficient discretizations for the multidimensional cases. 

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2. Efficient discretization and preconditioning of the singularly perturbed reaction-diffusion problem

   https://doi.org/https://doi.org/10.1016/j.camwa.2022.01.031  

   Constantin Bacuta; Daniel Hayes; Jacob Jacavage (January 2022, Computers mathematics with applications)

   Elsevier (Ed.)

   We consider the reaction diffusion problem and present efficient ways to discretize and precondition in the singular perturbed case when the reaction term dominates the equation. Using the concepts of optimal test norm and saddle point reformulation, we provide efficient discretization processes for uniform and non-uniform meshes. We present a preconditioning strategy that works for a large range of the perturbation parameter. Numerical examples to illustrate the efficiency of the method are included for a problem on the unit square. 

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3. Saddle point least squares discretization for convection-diffusion

   https://doi.org/10.1080/00036811.2023.2291511  

   Bacuta, Constantin; Hayes, Daniel; O'Grady, Tyler (August 2024, Applicable Analysis)

   We consider a model convection-diffusion problem and present our recent analysis and numerical results regarding mixed finite element formulation and discretization in the singular perturbed case when the convection term dominates the problem. Using the concepts of optimal norm and saddle point reformulation, we found new error estimates for the case of uniform meshes. We compare the standard linear Galerkin discretization to a saddle point least square discretization that uses quadratic test functions, and explain the non-physical oscillations of the discrete solutions. We also relate a known upwinding Petrov–Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. The results can be extended to the multidimensional case in order to find efficient approximations for more general singular perturbed problems including convection dominated models. 

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4. Uniform block-diagonal preconditioners for divergence-conforming HDG Methods for the generalized Stokes equations and the linear elasticity equations

   https://doi.org/10.1093/imanum/drac021  

   Fu, Guosheng; Kuang, Wenzheng (June 2022, IMA Journal of Numerical Analysis)

   Abstract We propose a uniform block-diagonal preconditioner for condensed $$H$$(div)-conforming hybridizable discontinuous Galerkin schemes for parameter-dependent saddle point problems, including the generalized Stokes equations and the linear elasticity equations. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning technique (Xu (1994) The Auxiliary Space Method and Optimal Multigrid Preconditioning Techniques for Unstructured Grids, vol. 56. International GAMM-Workshop on Multi-level Methods (Meisdorf), pp. 215–235). A spectrally equivalent approximation to the Schur complement on the element-wise constant pressure space is also constructed, and an explicit computable exact inverse is obtained via the Woodbury matrix identity. Finally, the numerical results verify the robustness of our proposed preconditioner with respect to model parameters and mesh size. 

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5. Transformed primal-dual methods for nonlinear saddle point systems

   https://doi.org/10.1515/jnma-2022-0056  

   Chen, Long; Wei, Jingrong (January 2023, Journal of Numerical Mathematics)

   Abstract A transformed primal-dual (TPD) flow is developed for a class of nonlinear smooth saddle point system. The flow for the dual variable contains a Schur complement which is strongly convex. Exponential stability of the saddle point is obtained by showing the strong Lyapunov property. Several TPD iterations are derived by implicit Euler, explicit Euler, implicit-explicit and Gauss-Seidel methods with accelerated overrelaxation of the TPD flow. Generalized to the symmetric TPD iterations, linear convergence rate is preserved for convex-concave saddle point systems under assumptions that the regularized functions are strongly convex. The effectiveness of augmented Lagrangian methods can be explained as a regularization of the non-strongly convexity and a preconditioning for the Schur complement. The algorithm and convergence analysis depends crucially on appropriate inner products of the spaces for the primal variable and dual variable. A clear convergence analysis with nonlinear inexact inner solvers is also developed. 

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