184 research outputs found

    hp-Version discontinuous Galerkin finite element methods for semilinear parabolic problems

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    We consider the hp-version interior penalty discontinuous Galerkin finite element method (hp-DGFEM) for semilinear parabolic equations with mixed Dirichlet and Neumann boundary conditions. Our main concern is the error analysis of the hp--DGFEM on shape--regular spatial meshes. We derive error bounds under various hypotheses on the regularity of the solution, for both the symmetric and non--symmetric versions of DGFEM

    Discontinuous Galerkin Methods for Mass Transfer through Semi-Permeable Membranes

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    A discontinuous Galerkin (dG) method for the numerical solution of initial/boundary value multi-compartment partial differential equation (PDE) models, interconnected with interface conditions, is presented and analysed. The study of interface problems is motivated by models of mass transfer of solutes through semi-permeable membranes. More specifically, a model problem consisting of a system of semilinear parabolic advection-diffusion-reaction partial differential equations in each compartment, equipped with respective initial and boundary conditions, is considered. Nonlinear interface conditions modelling selective permeability, congestion and partial reflection are applied to the compartment interfaces. An interior penalty dG method is presented for this problem and it is analysed in the space-discrete setting. The a priori analysis shows that the method yields optimal a priori bounds, provided the exact solution is sufficiently smooth. Numerical experiments indicate agreement with the theoretical bounds and highlight the stability of the numerical method in the advection-dominated regime

    One-parameter discontinuous Galerkin finite element discretisation of quasilinear parabolic problems

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    We consider the analysis of a one-parameter family of hphp--version discontinuous Galerkin finite element methods for the numerical solution of quasilinear parabolic equations of the form u'-\na\cdot\set{a(x,t,\abs{\na u})\na u}=f(x,t,u) on a bounded open set \om\in\re^d, subject to mixed Dirichlet and Neumann boundary conditions on \pr\om. It is assumed that aa is a real--valued function which is Lipschitz-continuous and uniformly monotonic in its last argument, and ff is a real-valued function which is locally Lipschitz-continuous and satisfies a suitable growth condition in its last argument; both functions are measurable in the first and second arguments. For quasi--uniform hphp--meshes, if u\in \H^1(0,T;\H^k(\om))\cap\L^\infty(0,T;\H^1(\om)) with kā‰„312k\geq 3\frac{1}{2}, for discontinuous piecewise polynomials of degree not less than 1, the approximation error, measured in the broken H1H^1 norm, is proved to be the same as in the linear case: O(hsāˆ’1/pkāˆ’3/2)\mathscr{O}(h^{s-1}/p^{k-3/2}) with 1ā‰¤sā‰¤minā”{ā€‰p+1,kā€‰}1\leq s\leq\min\set{p+1,k}

    An hp-version discontinuous Galerkin method for integro-differential equations of parabolic type

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    We study the numerical solution of a class of parabolic integro-differential equations with weakly singular kernels. We use an hphp-version discontinuous Galerkin (DG) method for the discretization in time. We derive optimal hphp-version error estimates and show that exponential rates of convergence can be achieved for solutions with singular (temporal) behavior near t=0t=0 caused by the weakly singular kernel. Moreover, we prove that by using nonuniformly refined time steps, optimal algebraic convergence rates can be achieved for the hh-version DG method. We then combine the DG time-stepping method with a standard finite element discretization in space, and present an optimal error analysis of the resulting fully discrete scheme. Our theoretical results are numerically validated in a series of test problems

    Localized solutions and filtering mechanisms for the discontinuous Galerkin semi-discretizations of the 1-d wave equation

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    We perform a complete Fourier analysis of the semi-discrete 1-d wave equation obtained through a P1 discontinuous Galerkin (DG) approximation of the continuous wave equation on an uniform grid. The resulting system exhibits the interaction of two types of components: a physical one and a spurious one, related to the possible discontinuities that the numerical solution allows. Each dispersion relation contains critical points where the corresponding group velocity vanishes. Following previous constructions, we rigorously build wave packets with arbitrarily small velocity of propagation concentrated either on the physical or on the spurious component. We also develop filtering mechanisms aimed at recovering the uniform velocity of propagation of the continuous solutions. Finally, some applications to numerical approximation issues of control problems are also presented.Comment: 6 pages, 2 figure
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