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In this paper we analyze a homogeneous parabolic problem with initial data in the space of regular Borel measures. The problem is discretized in time with a discontinuous Galerkin scheme of arbitrary degree and in space with continuous finite elements of orders one or two. We show parabolic smoothing results for the continuous, semidiscrete and fully discrete problems. Our main results are interior L∞ error estimates for the evaluation at the endtime, in cases where the initial data is supported in a subdomain. In order to obtain these, we additionally show interior L∞ error estimates for L2 initial data and quadratic finite elements, which extends the corresponding result previously established by the authors for linear finite elements. [ABSTRACT FROM AUTHOR]

The integral fractional Laplacian of order s ∈ (0, 1) is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity. This, in turn, deteriorates the global regularity of solutions and as a result the global convergence rate of the numerical solutions. For finite element discretizations, we derive local error estimates in the Hs-seminorm and show optimal convergence rates in the interior of the domain by only assuming meshes to be shape-regular. These estimates quantify the fact that the reduced approximation error is concentrated near the boundary of the domain. We illustrate our theoretical results with several numerical examples. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we establish a best approximation property of fully discrete Galerkin solutions of second-order parabolic problems on convex polygonal and polyhedral domains in the L∞(I;W1,∞(Ω)) norm. The discretization method consists of continuous Lagrange finite elements in space and discontinuous Galerkin methods of arbitrary order in time. The method of the proof differs from the established fully discrete error estimate techniques and uses only elliptic results and discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors [D. Leykekhman and B. Vexler, Numer. Math., 135 (2017), pp. 923-952]. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish an interior best approximation property that shows more local dependence of the error at a point. [ABSTRACT FROM AUTHOR]

In this paper we provide an a priori error analysis for parabolic optimal control problems with a pointwise (Dirac-type) control in space on three-dimensional domains. The twodimensional case was treated in [D. Leykekhman and B. Vexler, SIAM J. Numer. Anal., 51 (2013), pp. 2797-2821]; however, the three-dimensional case is technically much more involved. To approximate the problem we use standard continuous piecewise linear elements in space and the piecewise constant discontinuous Galerkin method in time. Despite low regularity of the state equation, we establish O(√k + h) order of convergence rate for the control in the L² norm. This result improves almost twice the previously known estimate in [W. Gong, M. Hinze, and Z. Zhou, SIAM J. Control Optim., 52 (2014), pp. 97-119] and in addition does not require any relationship between the time step k and the mesh size h. The main technical tools are discrete maximal parabolic regularity results and the best approximation-type estimate for the finite element error in the L∞(Ω; L²(I)) norm. [ABSTRACT FROM AUTHOR]

The main goal of the paper is to establish that the L1 norm of jumps of the normal derivative across element boundaries and the L1 norm of the Laplacian of a piecewise polynomial finite element function can be controlled by corresponding weighted discrete H2 norm on convex polyhedral domains. In the finite element literature such results are only available for piecewise linear elements in two dimensions and the extension to convex polyhedral domains is rather technical. As a consequence of this result, we establish almost pointwise stability of the Ritz projection and the discrete resolvent estimate in L∞ norm. [ABSTRACT FROM AUTHOR]

In this paper we consider a parabolic optimal control problem with a pointwise (Dirac type) control in space, but variable in time, in two space dimensions. To approximate the problem we use the standard continuous piecewise linear approximation in space and the piecewise constant discontinuous Galerkin method in time. Despite low regularity of the state equation, we show almost optimal h² + k convergence rate for the control in L² norm. This result improves almost twice the previously known estimate in [W. Gong, M. Hinze, and Z. Zhou, A Priori Error Analysis for Finite Element Approximation of Parabolic Optimal Control Problems with Pointwise Control, Tech. report, 2011-07, Hamburger Beiträge zur Angewandten Mathematik, Hamburg, Germany, 2011]. [ABSTRACT FROM AUTHOR]