278 research outputs found

    A Posteriori Error Estimation for Highly Indefinite Helmholtz Problems

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    We develop a new analysis for residual-type aposteriori error estimation for a class of highly indefinite elliptic boundary value problems by considering the Helmholtz equation at high wavenumber as our model problem. We employ a classical conforming Galerkin discretization by using hp-finite elements. In [Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp., 79 (2010), pp.1871-1914], Melenk and Sauter introduced an hp-finite element discretization which leads to a stable and pollution-free discretization of the Helmholtz equation under a mild resolution condition which requires only degrees of freedom, where denotes the spatial dimension. In the present paper, we will introduce an aposteriori error estimator for this problem and prove its reliability and efficiency. The constants in these estimates become independent of the, possibly, high wavenumber provided the aforementioned resolution condition for stability is satisfied. We emphasize that, by using the classical theory, the constants in the aposteriori estimates would be amplified by a factor

    A posteriori error estimation for highly indefinite Helmholtz problems

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    We develop a new analysis for residual-type a posteriori error estimation for a class of highly indefinite elliptic boundary value problems by considering the Helmholtz equation at high wavenumber k > 0 as our model problem. We employ a classical conforming Galerkin discretization by using hp-finite elements. In [Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions, Math. Comp., 79 (2010), pp. 1871-1914], Melenk and Sauter introduced an hp-finite element discretization which leads to a stable and pollution-free discretization of the Helmholtz equation under a mild resolution condition which requires only O(kd) degrees of freedom, where d = 1; 2; 3 denotes the spatial dimension. In the present paper, we will introduce an a posteriori error estimator for this problem and prove its reliability and efficiency. The constants in these estimates become independent of the, possibly, high wavenumber k > 0 provided the aforementioned resolution condition for stability is satisfied. We emphasize that, by using the classical theory, the constants in the a posteriori estimates would be amplified by a factor k

    Mini-Workshop: Efficient and Robust Approximation of the Helmholtz Equation

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    The accurate and efficient treatment of wave propogation phenomena is still a challenging problem. A prototypical equation is the Helmholtz equation at high wavenumbers. For this equation, Babuška & Sauter showed in 2000 in their seminal SIAM Review paper that standard discretizations must fail in the sense that the ratio of true error and best approximation error has to grow with the frequency. This has spurred the development of alternative, non-standard discretization techniques. This workshop focused on evaluating and comparing these different approaches also with a view to their applicability to more general wave propagation problems

    Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors

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    This paper addresses the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local cell problems is rigorously justified. The stabilization eliminates scale-dependent pre-asymptotic effects as they appear for standard finite element discretizations of highly oscillatory problems, e.g., the poor L2L^2 approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering

    Primal dual mixed finite element methods for indefinite advection--diffusion equations

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    We consider primal-dual mixed finite element methods for the advection--diffusion equation. For the primal variable we use standard continuous finite element space and for the flux we use the Raviart-Thomas space. We prove optimal a priori error estimates in the energy- and the L2L^2-norms for the primal variable in the low Peclet regime. In the high Peclet regime we also prove optimal error estimates for the primal variable in the H(div)H(div) norm for smooth solutions. Numerically we observe that the method eliminates the spurious oscillations close to interior layers that pollute the solution of the standard Galerkin method when the local Peclet number is high. This method, however, does produce spurious solutions when outflow boundary layer presents. In the last section we propose two simple strategies to remove such numerical artefacts caused by the outflow boundary layer and validate them numerically.Comment: 25 pages, 6 figures, 5 table

    Frequency-explicit a posteriori error estimates for discontinuous Galerkin discretizations of Maxwell's equations

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    We propose a new residual-based a posteriori error estimator for discontinuous Galerkin discretizations of time-harmonic Maxwell's equations in first-order form. We establish that the estimator is reliable and efficient, and the dependency of the reliability and efficiency constants on the frequency is analyzed and discussed. The proposed estimates generalize similar results previously obtained for the Helmholtz equation and conforming finite element discretization of Maxwell's equations. In addition, for the discontinuous Galerkin scheme considered here, we also show that the proposed estimator is asymptotically constant-free for smooth solutions. We also present two-dimensional numerical examples that highlight our key theoretical findings and suggest that the proposed estimator is suited to drive hh- and hphp-adaptive iterative refinements.Comment: arXiv admin note: substantial text overlap with arXiv:2009.0920
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