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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. Connections Between Finite Difference and Finite Element Approximations for a Convection-Diffusion Problem

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

   We consider a model convection-diffusion problem and present useful connections between the finite differences and finite element discretization methods. We introduce a general upwinding Petrov-Galerkin discretization based on bubble modification of the test space and connect the method with the general upwinding approach used in finite difference discretization. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices, and compare the right hand side load vectors for the two methods. This new approach allows for improving well known upwinding finite difference methods and for obtaining new error estimates. We prove that the exponential bubble Petrov-Galerkin discretization can recover the interpolant of the exact solution. As a consequence, we estimate the closeness of the related finite difference solutions to the interpolant. The ideas we present in this work, can lead to building efficient new discretization methods for multidimensional convection dominated problems. 

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4. An a priori error analysis of adjoint-based super-convergent Galerkin approximations of linear functionals

   https://doi.org/10.1093/imanum/draa102  

   Cockburn, Bernardo; Xia, Shiqiang (January 2021, IMA Journal of Numerical Analysis)

   Abstract We present the first a priori error analysis of a new method proposed in Cockburn & Wang (2017, Adjoint-based, superconvergent Galerkin approximations of linear functionals. J. Comput. Sci., 73, 644–666), for computing adjoint-based, super-convergent Galerkin approximations of linear functionals. If $J(u)$ is a smooth linear functional, where $$u$$ is the solution of a steady-state diffusion problem, the standard approximation $$J(u_h)$$ converges with order $$h^{2k+1}$$, where $$u_h$$ is the Hybridizable Discontinuous Galerkin approximation to $$u$$ with polynomials of degree $k>0$. In contrast, numerical experiments show that the new method provides an approximation that converges with order $$h^{4k}$$, and can be computed by only using twice the computational effort needed to compute $$J(u_h)$$. Here, we put these experimental results in firm mathematical ground. We also display numerical experiments devised to explore the convergence properties of the method in cases not covered by the theory, in particular, when the solution $$u$$ or the functional $$J(\cdot )$$ are not very smooth. We end by indicating how to extend these results to the case of general Galerkin methods. 

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5. A local discontinuous Galerkin method for nonlinear parabolic SPDEs

   https://doi.org/10.1051/m2an/2020026  

   Li, Yunzhang; Shu, Chi-Wang; Tang, Shanjian (January 2021, ESAIM: Mathematical Modelling and Numerical Analysis)

   null (Ed.)

   In this paper, we propose a local discontinuous Galerkin (LDG) method for nonlinear and possibly degenerate parabolic stochastic partial differential equations, which is a high-order numerical scheme. It extends the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and shares with the DG method its advantage and flexibility. We prove the L 2 -stability of the numerical scheme for fully nonlinear equations. Optimal error estimates ( O ( h (k+1) )) for smooth solutions of semi-linear stochastic equations is shown if polynomials of degree k are used. We use an explicit derivative-free order 1.5 time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples are given to display the performance of the LDG method. 

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