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  1. ; (, La Matematica)
    Abstract In this paper, we study the superconvergence of the semi-discrete discontinuous Galerkin (DG) method for linear hyperbolic equations in one spatial dimension. The asymptotic errors in cell averages, downwind point values, and the postprocessed solution are derived for the initial discretization by Gaussian projection (for periodic boundary condition) or Cao projection Cao et al. (SIAM J. Numer. Anal.5, 2555–2573 (2014)) (for Dirichlet boundary condition). We proved that the error constant in the superconvergence of order$$2k+1$$ 2 k + 1 for DG methods based on upwind-biased fluxes depends on the parity of the orderk. The asymptotic errors are demonstrated by various numerical experiments for scalar and vector hyperbolic equations. 
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  2. (, Statistics & Probability Letters)
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  3. ; ; (, Journal of Computational Physics)
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  4. ; (, Journal of Differential Equations)
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  5. ; (, Journal of Inverse and Ill-posed Problems)
    Abstract The inverse problem in gravimetry is to find a domain 𝐷 inside the reference domain Ω from boundary measurements of gravitational force outside Ω.We found that about five parameters of the unknown 𝐷 can be stably determined given data noise in practical situations.An ellipse is uniquely determined by five parameters.We prove uniqueness and stability of recovering an ellipse for the inverse problem from minimal amount of data which are the gravitational force at three boundary points.In the proofs, we derive and use simple systems of linear and nonlinear algebraic equations for natural parameters of an ellipse.To illustrate the technique, we use these equations in numerical examples with various location of measurements points on ∂ ⁡ Ω \partial\Omega .Similarly, a rectangular 𝐷 is considered.We consider the problem in the plane as a model for the three-dimensional problem due to simplicity. 
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