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1. Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source

   https://doi.org/10.1051/m2an/2022095  

   Masri, Rami; Shen, Boqian; Riviere, Beatrice (March 2023, ESAIM: Mathematical Modelling and Numerical Analysis)

   The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a priori error estimates in the L 2 norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the L 2 norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the L 2 norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in L 2 in time and in space. Numerical results for the elliptic problem are added to support the theoretical results. 

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2. A mixed elasticity formulation for fluid–poroelastic structure interaction

   https://doi.org/10.1051/m2an/2021083  

   Li, Tongtong; Yotov, Ivan (January 2022, ESAIM: Mathematical Modelling and Numerical Analysis)

   We develop a mixed finite element method for the coupled problem arising in the interaction between a free fluid governed by the Stokes equations and flow in deformable porous medium modeled by the Biot system of poroelasticity. Mass conservation, balance of stress, and the Beavers–Joseph–Saffman condition are imposed on the interface. We consider a fully mixed Biot formulation based on a weakly symmetric stress-displacement-rotation elasticity system and Darcy velocity-pressure flow formulation. A velocity-pressure formulation is used for the Stokes equations. The interface conditions are incorporated through the introduction of the traces of the structure velocity and the Darcy pressure as Lagrange multipliers. Existence and uniqueness of a solution are established for the continuous weak formulation. Stability and error estimates are derived for the semi-discrete continuous-in-time mixed finite element approximation. Numerical experiments are presented to verify the theoretical results and illustrate the robustness of the method with respect to the physical parameters. 

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3. Mathematical and numerical analysis for PDE systems modeling intravascular drug release from arterial stents and transport in arterial tissue

   https://doi.org/10.3934/mbe.2024248  

   Feng, Xiaobing; Jiang, Tingao (January 2024, Mathematical Biosciences and Engineering)

   This paper is concerned with the PDE (partial differential equation) and numerical analysis of a modified one-dimensional intravascular stent model. It is proved that the modified model has a unique weak solution by using the Galerkin method combined with a compactness argument. A semi-discrete finite-element method and a fully discrete scheme using the Euler time-stepping have been formulated for the PDE model. Optimal order error estimates in the energy norm are proved for both schemes. Numerical results are presented, along with comparisons between different decoupling strategies and time-stepping schemes. Lastly, extensions of the model and its PDE and numerical analysis results to the two-dimensional case are also briefly discussed. 

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4. A cutFEM divergence–free discretization for the stokes problem

   https://doi.org/10.1051/m2an/2022072  

   Liu, Haoran; Neilan, Michael; Olshanskii, Maxim (January 2023, ESAIM: Mathematical Modelling and Numerical Analysis)

   We construct and analyze a CutFEM discretization for the Stokes problem based on the Scott–Vogelius pair. The discrete piecewise polynomial spaces are defined on macro-element triangulations which are not fitted to the smooth physical domain. Boundary conditions are imposed via penalization through the help of a Nitsche-type discretization, whereas stability with respect to small and anisotropic cuts of the bulk elements is ensured by adding local ghost penalty stabilization terms. We show stability of the scheme as well as a divergence–free property of the discrete velocity outside an O ( h ) neighborhood of the boundary. To mitigate the error caused by the violation of the divergence–free condition, we introduce local grad–div stabilization. The error analysis shows that the grad–div parameter can scale like O ( h −1 ), allowing a rather heavy penalty for the violation of mass conservation, while still ensuring optimal order error estimates. 

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5. A mass- and energy-conserved DG method for the Schrödinger-Poisson equation

   https://doi.org/10.1007/s11075-021-01139-0  

   Yi, Nianyu; Liu, Hailiang (February 2021, Numerical Algorithms)

   null (Ed.)

   In this paper, we construct, analyze, and numerically validate conservative discontinuous Galerkin (DG) schemes for approximating the Schr\"{o}dinger-Poisson equation. The proposed schemes all satisfy both mass and energy conservation. For the semi-discrete DG scheme optimal $L^2$ error estimates are obtained. Efficient iterative solvers are also constructed to solve the second order implicit time discretization. A number of numerical tests are presented to demonstrate the method’s accuracy and robustness, confirming that both mass and energy are well preserved over long time simulations. 

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