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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. 

This paper is concerned with developing an efficient numerical algorithm for the fast implementation of the sparse grid method for computing the d-dimensional integral of a given function. The new algorithm, called the MDI-SG (multilevel dimension iteration sparse grid) method, implements the sparse grid method based on a dimension iteration/reduction procedure. It does not need to store the integration points, nor does it compute the function values independently at each integration point; instead, it reuses the computation for function evaluations as much as possible by performing the function evaluations at all integration points in a cluster and iteratively along coordinate directions. It is shown numerically that the computational complexity (in terms of CPU time) of the proposed MDI-SG method is of polynomial order O(d3Nb)(b≤2) or better, compared to the exponential order O(N(logN)d−1) for the standard sparse grid method, where N denotes the maximum number of integration points in each coordinate direction. As a result, the proposed MDI-SG method effectively circumvents the curse of dimensionality suffered by the standard sparse grid method for high-dimensional numerical integration. 

This article develops a unified general framework for designing convergent finite difference and discontinuous Galerkin methods for approximating viscosity and regular solutions of fully nonlinear second order PDEs. Unlike the well-known monotone (finite difference) framework, the proposed new framework allows for the use of narrow stencils and unstructured grids which makes it possible to construct high order methods. The general framework is based on the concepts of consistency and g-monotonicity which are both defined in terms of various numerical derivative operators. Specific methods that satisfy the framework are constructed using numerical moments. Admissibility, stability, and convergence properties are proved,and numerical experiments are provided along with some computer implementation details. For more information see https://ejde.math.txstate.edu/conf-proc/26/f1/abstr.html 

Abstract In this paper, a higher order time-discretization scheme is proposed, where the iterates approximate the solution of the stochastic semilinear wave equation driven by multiplicative noise with general drift and diffusion. We employ variational method for its error analysis and prove an improved convergence order of $$\frac 32$$ for the approximates of the solution. The core of the analysis is Hölder continuity in time and moment bounds for the solutions of the continuous and the discrete problem. Computational experiments are also presented.