87 research outputs found
Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations
We consider the convergence theory of adaptive multigrid methods for second-order elliptic problems and Maxwell's equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their "immediate” neighbors. In the context of lowest order conforming finite element approximations, we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms. The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures. The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom. We demonstrate our convergence theory by two numerical experiment
A fourth-order unfitted characteristic finite element method for solving the advection-diffusion equation on time-varying domains
We propose a fourth-order unfitted characteristic finite element method to
solve the advection-diffusion equation on time-varying domains. Based on a
characteristic-Galerkin formulation, our method combines the cubic MARS method
for interface tracking, the fourth-order backward differentiation formula for
temporal integration, and an unfitted finite element method for spatial
discretization. Our convergence analysis includes errors of discretely
representing the moving boundary, tracing boundary markers, and the spatial
discretization and the temporal integration of the governing equation.
Numerical experiments are performed on a rotating domain and a severely
deformed domain to verify our theoretical results and to demonstrate the
optimal convergence of the proposed method
A PML method for signal-propagation problems in axon
This work is focused on the modelling of signal propagations in myelinated
axons to characterize the functions of the myelin sheath in the neural
structure. Based on reasonable assumptions on the medium properties, we derive
a two-dimensional neural-signaling model in cylindrical coordinates from the
time-harmonic Maxwell's equations. The well-posedness of model is established
upon Dirichlet boundary conditions at the two ends of the neural structure and
the radiative condition in the radial direction of the structure. Using the
perfectly matched layer (PML) method, we truncate the unbounded background
medium and propose an approximate problem on the truncated domain. The
well-posedness of the PML problem and the exponential convergence of the
approximate solution to the exact solution are established. Numerical
experiments based on finite element discretization are presented to demonstrate
the theoretical results and the efficiency of our methods to simulate the
signal propagation in axons
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