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1. Numerical analysis of sparse initial data identification for parabolic problems

   https://doi.org/10.1051/m2an/2019083  

   Leykekhman, Dmitriy; Vexler, Boris; Walter, Daniel (July 2020, ESAIM: Mathematical Modelling and Numerical Analysis)

   null (Ed.)

   In this paper we consider a problem of initial data identification from the final time observation for homogeneous parabolic problems. It is well-known that such problems are exponentially ill-posed due to the strong smoothing property of parabolic equations. We are interested in a situation when the initial data we intend to recover is known to be sparse, i.e. its support has Lebesgue measure zero. We formulate the problem as an optimal control problem and incorporate the information on the sparsity of the unknown initial data into the structure of the objective functional. In particular, we are looking for the control variable in the space of regular Borel measures and use the corresponding norm as a regularization term in the objective functional. This leads to a convex but non-smooth optimization problem. For the discretization we use continuous piecewise linear finite elements in space and discontinuous Galerkin finite elements of arbitrary degree in time. For the general case we establish error estimates for the state variable. Under a certain structural assumption, we show that the control variable consists of a finite linear combination of Dirac measures. For this case we obtain error estimates for the locations of Dirac measures as well as for the corresponding coefficients. The key to the numerical analysis are the sharp smoothing type pointwise finite element error estimates for homogeneous parabolic problems, which are of independent interest. Moreover, we discuss an efficient algorithmic approach to the problem and show several numerical experiments illustrating our theoretical results. 

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2. A C ⁰ interior penalty method for the stream function formulation of the surface Stokes problem

   https://doi.org/10.1051/m2an/2025025  

   Neilan, Michael; Wan, Hongzhi (March 2025, ESAIM: Mathematical Modelling and Numerical Analysis)

   We propose aC⁰interior penalty method for the fourth-order stream function formulation of the surface Stokes problem. The scheme utilizes continuous, piecewise polynomial spaces defined on an approximate surface. We show that the resulting discretization is positive definite and derive error estimates in various norms in terms of the polynomial degree of the finite element space as well as the polynomial degree to define the geometry approximation. A notable feature of the scheme is that it does not explicitly depend on the Gauss curvature of the surface. This is achievedviaa novel integration-by-parts formula for the surface biharmonic operator. 

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3. 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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4. An H(div)-conforming Finite Element Method for Biot’s Consolidation Model

   Zeng, Yuping; Cai, Mingchao; Wang, Feng. (August 2019, East Asian Journal on Applied Mathematics)

   An H(div)-conforming finite element method for the Biot’s consolidation mo- del is developed, with displacements and fluid velocity approximated by elements from BDM_k space. The use of H(div)-conforming elements for flow variables ensures the local mass conservation. In the H(div)-conforming approximation of displacement, the tan- gential components are discretised in the interior penalty discontinuous Galerkin frame- work,and the normal components across the element interfaces are continuous. Having introduced a spatial discretisation, we develop a semi-discrete scheme and a fully dis- crete scheme,prove their unique solvability and establish optimal error estimates for each variable. 

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5. $$L^2(I;H^1(\Omega)^d)$$ and $$L^2(I;L^2(\Omega)^d)$$ best approximation type error estimates for Galerkin solutions of transient Stokes problems

   Leykekhman, Dmitriy; Vexler, Boris (October 2023, Calcolo)

   In this paper we establish best approximation type estimates for the fully discrete Galerkin solutions of transient Stokes problem in $$L^2(I;L^2(\Omega)^d)$$ and $$L^2(I;H^1(\Omega)^d)$$ norms. These estimates fill the gap in the error analysis of the transient Stokes problems and have a number of applications. The analysis naturally extends to inhomogeneous parabolic problems. The best type $$L^2(I;H^1(\Omega))$$ error estimate are new even for scalar parabolic problems. 

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