158 research outputs found
Multigrid methods for Maxwell\u27s equations
In this work we study finite element methods for two-dimensional Maxwell\u27s equations and their solutions by multigrid algorithms. We begin with a brief survey of finite element methods for Maxwell\u27s equations. Then we review the related fundamentals, such as Sobolev spaces, elliptic regularity results, graded meshes, finite element methods for second order problems, and multigrid algorithms. In Chapter 3, we study two types of nonconforming finite element methods on graded meshes for a two-dimensional curl-curl and grad-div problem that appears in electromagnetics. The first method is based on a discretization using weakly continuous P1 vector fields. The second method uses discontinuous P1 vector fields. Optimal convergence rates (up to an arbitrary positive epsilon) in the energy norm and the L2 norm are established for both methods on graded meshes. In Chapter 4, we consider a class of symmetric discontinuous Galerkin methods for a model Poisson problem on graded meshes that share many techniques with the nonconforming methods in Chapter 3. Optimal order error estimates are derived in both the energy norm and the L2 norm. Then we establish the uniform convergence of W-cycle, V-cycle and F-cycle multigrid algorithms for the resulting discrete problems. In Chapter 5, we propose a new numerical approach for two-dimensional Maxwell\u27s equations that is based on the Hodge decomposition for divergence-free vector fields. In this approach, an approximate solution for Maxwell\u27s equations can be obtained by solving standard second order scalar elliptic boundary value problems. We illustrate this new approach by a P1 finite element method. In Chapter 6, we first report numerical results for multigrid algorithms applied to the discretized curl-curl and grad-div problem using nonconforming finite element methods. Then we present multigrid results for Maxwell\u27s equations based on the approach introduced in Chapter 5. All the theoretical results obtained in this dissertation are confirmed by numerical experiments
Stable finite element pair for Stokes problem and discrete Stokes complex on quadrilateral grids
In this paper, we first construct a nonconforming finite element pair for the
incompressible Stokes problem on quadrilateral grids, and then construct a
discrete Stokes complex associated with that finite element pair. The finite
element spaces involved consist of piecewise polynomials only, and the
divergence-free condition is imposed in a primal formulation. Combined with
some existing results, these constructions can be generated onto grids that
consist of both triangular and quadrilateral cells
Nonconforming finite element Stokes complexes in three dimensions
Two nonconforming finite element Stokes complexes ended with the
nonconforming - element for the Stokes equation in three dimensions
are constructed. And commutative diagrams are also shown by combining
nonconforming finite element Stokes complexes and interpolation operators. The
lower order -nonconforming finite
element only has degrees of freedom, whose basis functions are explicitly
given in terms of the barycentric coordinates. The -nonconforming elements are applied to solve the
quad-curl problem, and optimal convergence is derived. By the nonconforming
finite element Stokes complexes, the mixed finite element methods of the
quad-curl problem is decoupled into two mixed methods of the Maxwell equation
and the nonconforming - element method for the Stokes equation, based
on which a fast solver is developed.Comment: 20 page
Genetic Exponentially Fitted Method for Solving Multi-dimensional Drift-diffusion Equations
A general approach was proposed in this article to develop high-order
exponentially fitted basis functions for finite element approximations of
multi-dimensional drift-diffusion equations for modeling biomolecular
electrodiffusion processes. Such methods are highly desirable for achieving
numerical stability and efficiency. We found that by utilizing the one-one
correspondence between continuous piecewise polynomial space of degree
and the divergence-free vector space of degree , one can construct
high-order 2-D exponentially fitted basis functions that are strictly
interpolative at a selected node set but are discontinuous on edges in general,
spanning nonconforming finite element spaces. First order convergence was
proved for the methods constructed from divergence-free Raviart-Thomas space
at two different node set
A convergent nonconforming finite element method for compressible Stokes flow
We propose a nonconforming finite element method for isentropic viscous gas
flow in situations where convective effects may be neglected. We approximate
the continuity equation by a piecewise constant discontinuous Galerkin method.
The velocity (momentum) equation is approximated by a finite element method on
div-curl form using the nonconforming Crouzeix-Raviart space. Our main result
is that the finite element method converges to a weak solution. The main
challenge is to demonstrate the strong convergence of the density
approximations, which is mandatory in view of the nonlinear pressure function.
The analysis makes use of a higher integrability estimate on the density
approximations, an equation for the "effective viscous flux", and renormalized
versions of the discontinuous Galerkin method.Comment: 23 page
Superconvergence of a nonconforming brick element for the quad-curl problem
This short note shows the superconvergence of an
-nonconforming brick element very recently
introduced in [17] for the quad-curl problem. The supercloseness is based on
proper modifications for both the interpolation and the discrete formulation,
leading to an superclose order in the discrete
norm. Moreover, we propose a suitable
postprocessing method to ensure the global superconvergence. Numerical results
verify our theory
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