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  • 1
    Publication Date: 2022-01-14
    Description: We study consistency of cell-centered finite difference methods for elliptic equations with degenerate coefficients in any space dimension $d \geq 2$. This results in order of convergence estimates in the natural weighted energy norm and in the weighted discrete $L^2$-norm on admissible meshes. The cells of meshes under consideration may be very irregular in size. We particularly allow the size of certain cells to remain bounded from below even in the asymptotic limit. For uniform meshes we show that the order of convergence is at least 1 in the energy semi-norm, provided the discrete and continuous solutions exist and the continuous solution has $H^2$ regularity.
    Language: English
    Type: article , doc-type:article
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  • 2
    Publication Date: 2019-01-30
    Description: In recent years, for the analysis of molecular processes, the estimation of time-scales and transition rates has become fundamental. Estimating the transition rates between molecular conformations is—from a mathematical point of view—an invariant subspace projection problem. We present a method to project the infinitesimal generator acting on function space to a low-dimensional rate matrix. This projection can be performed in two steps. First, we discretize the conformational space in a Voronoi tessellation, then the transition rates between adjacent cells is approximated by the geometric average of the Boltzmann weights of the Voronoi cells. This method demonstrates that there is a direct relation between the potential energy surface of molecular structures and the transition rates of conformational changes. We will show also that this approximation is correct and converges to the generator of the Smoluchowski equation in the limit of infinitely small Voronoi cells. We present results for a two dimensional diffusion process and alanine dipeptide as a high-dimensional system.
    Language: English
    Type: article , doc-type:article
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