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Register a new userA controllability method for Maxwells equations
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Added by
Th\\'eophile Chaumont-Frelet
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We propose a controllability method for the numerical solution of time-harmonic Maxwells equations in their first-order formulation. By minimizing a quadratic cost functional, which measures the deviation from periodicity, the controllability method determines iteratively a periodic solution in the time domain. At each conjugate gradient iteration, the gradient of the cost functional is simply computed by running any time-dependent simulation code forward and backward for one period, thus leading to a non-intrusive implementation easily integrated into existing software. Moreover, the proposed algorithm automatically inherits the parallelism, scalability, and low memory footprint of the underlying time-domain solver. Since the time-periodic solution obtained by minimization is not necessarily unique, we apply a cheap post-processing filtering procedure which recovers the time-harmonic solution from any minimizer. Finally, we present a series of numerical examples which show that our algorithm greatly speeds up the convergence towards the desired time-harmonic solution when compared to simply running the time-marching code until the time-harmonic regime is eventually reached.
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This work is devoted to the development and analysis of a linearization algorithm for microscopic elliptic equations, with scaled degenerate production, posed in a perforated medium and constrained by the homogeneous Neumann-Dirichlet boundary conditions. This technique plays two roles: to guarantee the unique weak solvability of the microscopic problem and to provide a fine approximation in the macroscopic setting. The scheme systematically relies on the choice of a stabilization parameter in such a way as to guarantee the strong convergence in $H^1$ norm for both the microscopic and macroscopic problems. In the standard variational setting, we prove the $H^1$-type contraction at the micro-scale based on the energy method. Meanwhile, we adopt the classical homogenization result in line with corrector estimate to show the convergence of the scheme at the macro-scale. In the numerical section, we use the standard finite element method to assess the efficiency and convergence of our proposed algorithm.
We propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility method with a suitable Carleman weight function. The presence of the Carleman weight function allows us to employ a Carleman estimate to prove the convergence of the sequence generated by the iterative scheme above to the desired solution. The convergence of the iteration is fast at an exponential rate without the need of an initial good guess. We apply this method to compute solutions to some general quasilinear elliptic equations and a large class of first-order Hamilton-Jacobi equations. Numerical results are presented.
High-precision numerical scheme for nonlinear hyperbolic evolution equations is proposed based on the spectral method. The detail discretization processes are discussed in case of one-dimensional Klein-Gordon equations. In conclusion, a numerical scheme with the order of total calculation cost $O(N log 2N)$ is proposed. As benchmark results, the relation between the numerical precision and the discretization unit size are demonstrated.
It is well known that, with a particular choice of norm, the classical double-layer potential operator $D$ has essential norm $<1/2$ as an operator on the natural trace space $H^{1/2}(Gamma)$ whenever $Gamma$ is the boundary of a bounded Lipschitz domain. This implies, for the standard second-kind boundary integral equations for the interior and exterior Dirichlet and Neumann problems in potential theory, convergence of the Galerkin method in $H^{1/2}(Gamma)$ for any sequence of finite-dimensional subspaces $(mathcal{H}_N)_{N=1}^infty$ that is asymptotically dense in $H^{1/2}(Gamma)$. Long-standing open questions are whether the essential norm is also $<1/2$ for $D$ as an operator on $L^2(Gamma)$ for all Lipschitz $Gamma$ in 2-d; or whether, for all Lipschitz $Gamma$ in 2-d and 3-d, or at least for the smaller class of Lipschitz polyhedra in 3-d, the weaker condition holds that the operators $pm frac{1}{2}I+D$ are compact perturbations of coercive operators -- this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspaces $(mathcal{H}_N)_{N=1}^infty$ that is asymptotically dense in $L^2(Gamma)$. We settle these open questions negatively. We give examples of 2-d and 3-d Lipschitz domains with Lipschitz constant equal to one for which the essential norm of $D$ is $geq 1/2$, and examples with Lipschitz constant two for which the operators $pm frac{1}{2}I +D$ are not coercive plus compact. We also give, for every $C>0$, examples of Lipschitz polyhedra for which the essential norm is $geq C$ and for which $lambda I+D$ is not a compact perturbation of a coercive operator for any real or complex $lambda$ with $|lambda|leq C$. Finally, we resolve negatively a related open question in the convergence theory for collocation methods.
A new finite difference method on irregular, locally perturbed rectangular grids has been developed for solving electromagnetic waves around curved perfect electric conductors (PEC). This method incorporates the back and forth error compensation and correction method (BFECC) and level set method to achieve convenience and higher order of accuracy at complicated PEC boundaries. A PDE-based local second order ghost cell extension technique is developed based on the level set framework in order to compute the boundary value to first order accuracy (cumulatively), and then BFECC is applied to further improve the accuracy while increasing the CFL number. Numerical experiments are conducted to validate the properties of the method.
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