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1. A Banach space mixed formulation for the unsteady Brinkman–Forchheimer equations

   https://doi.org/10.1093/imanum/draa035  

   Caucao, Sergio; Yotov, Ivan (August 2020, IMA Journal of Numerical Analysis)

   null (Ed.)

   Abstract We propose and analyse a mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Our approach is based on the introduction of a pseudostress tensor related to the velocity gradient and pressure, leading to a mixed formulation where the pseudostress tensor and the velocity are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators and a regularization technique. We then present well posedness and error analysis for semidiscrete continuous-in-time and fully discrete finite element approximations on simplicial grids with spatial discretization based on the Raviart–Thomas spaces of degree $$k$$ for the pseudostress tensor and discontinuous piecewise polynomial elements of degree $$k$$ for the velocity and backward Euler time discretization. We provide several numerical results to confirm the theoretical rates of convergence and illustrate the performance and flexibility of the method for a range of model parameters. 

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2. Optimal error estimates of the semidiscrete discontinuous Galerkin methods for two dimensional hyperbolic equations on Cartesian meshes using P ^k elements

   https://doi.org/10.1051/m2an/2019080  

   Liu, Yong; Shu, Chi-Wang; Zhang, Mengping (March 2020, ESAIM: Mathematical Modelling and Numerical Analysis)

   In this paper, we study the optimal error estimates of the classical discontinuous Galerkin method for time-dependent 2-D hyperbolic equations using P k elements on uniform Cartesian meshes, and prove that the error in the L 2 norm achieves optimal ( k  + 1)th order convergence when upwind fluxes are used. For the linear constant coefficient case, the results hold true for arbitrary piecewise polynomials of degree k  ≥ 0. For variable coefficient and nonlinear cases, we give the proof for piecewise polynomials of degree k  = 0, 1, 2, 3 and k  = 2, 3, respectively, under the condition that the wind direction does not change. The theoretical results are verified by numerical examples. 

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3. Stability analysis and error estimates of arbitrary Lagrangian–Eulerian discontinuous Galerkin method coupled with Runge–Kutta time-marching for linear conservation laws

   https://doi.org/10.1051/m2an/2018069  

   Zhou, Lingling; Xia, Yinhua; Shu, Chi-Wang (January 2019, ESAIM: Mathematical Modelling and Numerical Analysis)

   In this paper, we discuss the stability and error estimates of the fully discrete schemes for linear conservation laws, which consists of an arbitrary Lagrangian–Eulerian discontinuous Galerkin method in space and explicit total variation diminishing Runge–Kutta (TVD-RK) methods up to third order accuracy in time. The scaling arguments and the standard energy analysis are the key techniques used in our work. We present a rigorous proof to obtain stability for the three fully discrete schemes under suitable CFL conditions. With the help of the reference cell, the error equations are easy to establish and we derive the quasi-optimal error estimates in space and optimal convergence rates in time. For the Euler-forward scheme with piecewise constant elements, the second order TVD-RK method with piecewise linear elements and the third order TVD-RK scheme with polynomials of any order, the usual CFL condition is required, while for other cases, stronger time step restrictions are needed for the results to hold true. More precisely, the Euler-forward scheme needs τ ≤ ρh 2 and the second order TVD-RK scheme needs $$ \tau \le \rho {h}^{\frac{4}{3}}$$ for higher order polynomials in space, where τ and h are the time and maximum space step, respectively, and ρ is a positive constant independent of τ and h . 

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4. Construction of Supplemental Functions for Direct Serendipity and Mixed Finite Elements on Polygons

   https://doi.org/10.3390/math11224663  

   Arbogast, Todd; Wang, Chuning (November 2023, Mathematics)

   New families of direct serendipity and direct mixed finite elements on general planar, strictly convex polygons were recently defined by the authors. The finite elements of index r are H1 and H(div) conforming, respectively, and approximate optimally to order r+1 while using the minimal number of degrees of freedom. The shape function space consists of the full set of polynomials defined directly on the element and augmented with a space of supplemental functions. The supplemental functions were constructed as rational functions, which can be difficult to integrate accurately using numerical quadrature rules when the index is high. This can result in a loss of accuracy in certain cases. In this work, we propose alternative ways to construct the supplemental functions on the element as continuous piecewise polynomials. One approach results in supplemental functions that are in Hp for any p≥1. We prove the optimal approximation property for these new finite elements. We also perform numerical tests on them, comparing results for the original supplemental functions and the various alternatives. The new piecewise polynomial supplements can be integrated accurately, and therefore show better robustness with respect to the underlying meshes used. 

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5. A Computational Framework for the Numerical Solution of Optimal Control Problems Governed by Partial Differential Equations

   Davies, A M; Dennis, M E; Rao, A V (July 2024, American Control Conference)

   A computational framework for the solution of op- timal control problems with time-dependent partial differential equations (PDEs) is presented. The optimal control problem is transformed from a continuous time and space optimal control problem to a sparse nonlinear programming problem through state parameterization with Lagrange polynomials and discrete controls defined at Legendre-Gauss-Radau (LGR) points. The standard LGR collocation method is coupled with a modified Radau method to produce a collocation point on the typically noncollocated boundary. The newly collocated endpoint allows for a representation of the state derivative and control on the originally noncollocated boundary such that Neumann boundary conditions may be satisfied. Finally, the method developed in this paper is demonstrated on a viscous Burgers’ tracking problem and the results are compared to an existing solution. 

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