Anders Szepessy

CONVERGENCE OF THE DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR HYPERBOLIC CONSERVATION LAWS

We prove convergence of the discontinuous Galerkin finite element method with polynomials of arbitrary degree q≥0 on general unstructured meshes for scalar conservation laws in multidimensions. We also prove for systems of conservation laws that limits of discontinuous Galerkin finite element solutions satisfy the entropy inequalities of the system related to convex entropies.

Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions

Convergence of a shock-capturing streamline diffusion finite element method for a conservation law in two space dimensions

Stability of rarefaction waves in viscous media

AbstractWe study the time-asymptotic behavior of weak rarefaction waves of systems of conservation laws describing one-dimensional viscous media, with strictly hyperbolic flux functions. Our main result is to show that solutions of perturbed rarefaction data converge to an approximate, “Burgers” rarefaction wave, for initial perturbations w0 with small mass and localized as w0(x)=

Adaptive Multilevel Monte Carlo Simulation

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Adaptive Monte Carlo Algorithms for Stopped Diffusion

The proof proceeds by iteration of a pointwise ansatz for the error, using integral representations of its various components, based on Greens functions. We estimate the Greens functions by careful use of the Hopf-Cole transformation, combined with a refined parametrix method. As a consequence of our method, we also obtain rates of decay and detailed pointwise estimates for the error.This pointwise method has been used successfully in studying stability of shock and constant-state solutions. New features in the rarefaction case are time-varying coefficients in the linearized equations and error waves of unbounded mass

A remark on the stability of viscous shock waves

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