43 research outputs found
Towards finite element exterior calculus on manifolds: commuting projections, geometric variational crimes, and approximation errors
We survey recent contributions to finite element exterior calculus on
manifolds and surfaces within a comprehensive formalism for the error analysis
of vector-valued partial differential equations on manifolds. Our primary focus
is on uniformly bounded commuting projections on manifolds: these projections
map from Sobolev de Rham complexes onto finite element de Rham complexes,
commute with the differential operators, and satisfy uniform bounds in Lebesgue
norms. They enable the Galerkin theory of Hilbert complexes for a large range
of intrinsic finite element methods on manifolds. However, these intrinsic
finite element methods are generally not computable and thus primarily of
theoretical interest. This leads to our second point: estimating the geometric
variational crime incurred by transitioning to computable approximate problems.
Lastly, our third point addresses how to estimate the approximation error of
the intrinsic finite element method in terms of the mesh size. If the solution
is not continuous, then such an estimate is achieved via modified Cl\'ement or
Scott-Zhang interpolants that facilitate a broken Bramble--Hilbert lemma.Comment: Contribution to ENUMATH Proceedings 2023. 8 page
An odyssey into local refinement and multilevel preconditioning III: Implementation and numerical experiments
In this paper, we examine a number of additive and multiplicative multilevel iterative methods and preconditioners in the setting of two-dimensional local mesh refinement. While standard multilevel methods are effective for uniform refinement-based discretizations of elliptic equations, they tend to be less effective for algebraic systems, which arise from discretizations on locally refined meshes, losing their optimal behavior in both storage and computational complexity. Our primary focus here is on Bramble, Pasciak, and Xu (BPX)-style additive and multiplicative multilevel preconditioners, and on various stabilizations of the additive and multiplicative hierarchical basis (HB) method, and their use in the local mesh refinement setting. In parts I and II of this trilogy, it was shown that both BPX and wavelet stabilizations of HB have uniformly bounded condition numbers on several classes of locally refined two- and three-dimensional meshes based on fairly standard (and easily implementable) red and red-green mesh refinement algorithms. In this third part of the trilogy, we describe in detail the implementation of these types of algorithms, including detailed discussions of the data structures and traversal algorithms we employ for obtaining optimal storage and computational complexity in our implementations. We show how each of the algorithms can be implemented using standard data types, available in languages such as C and FORTRAN, so that the resulting algorithms have optimal (linear) storage requirements, and so that the resulting multilevel method or preconditioner can be applied with optimal (linear) computational costs. We have successfully used these data structure ideas for both MATLAB and C implementations using the FEtk, an open source finite element software package. We finish the paper with a sequence of numerical experiments illustrating the effectiveness of a number of BPX and stabilized HB variants for several examples requiring local refinement
Multigrid methods for discrete elliptic problems on triangular surfaces
We construct and analyze multigrid methods for discretized self-adjoint elliptic problems on triangular surfaces in R3. The methods involve the same weights for restriction and prolongation as in the case of planar triangulations and therefore are easy to implement. We prove logarithmic bounds of the convergence rates with constants solely depending on the ellipticity, the smoothers and on the regularity of the triangles forming the triangular surface. Our theoretical results are illustrated by numerical computations
An improved method for solving quasilinear convection diffusion problems on a coarse mesh
A method is developed for solving quasilinear convection diffusion problems
starting on a coarse mesh where the data and solution-dependent coefficients
are unresolved, the problem is unstable and approximation properties do not
hold. The Newton-like iterations of the solver are based on the framework of
regularized pseudo-transient continuation where the proposed time integrator is
a variation on the Newmark strategy, designed to introduce controllable
numerical dissipation and to reduce the fluctuation between the iterates in the
coarse mesh regime where the data is rough and the linearized problems are
badly conditioned and possibly indefinite. An algorithm and updated marking
strategy is presented to produce a stable sequence of iterates as boundary and
internal layers in the data are captured by adaptive mesh partitioning. The
method is suitable for use in an adaptive framework making use of local error
indicators to determine mesh refinement and targeted regularization. Derivation
and q-linear local convergence of the method is established, and numerical
examples demonstrate the theory including the predicted rate of convergence of
the iterations.Comment: 21 pages, 8 figures, 1 tabl
Multilevel Solvers for Unstructured Surface Meshes
Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner
Semilinear mixed problems on Hilbert complexes and their numerical approximation
Arnold, Falk, and Winther recently showed [Bull. Amer. Math. Soc. 47 (2010),
281-354] that linear, mixed variational problems, and their numerical
approximation by mixed finite element methods, can be studied using the
powerful, abstract language of Hilbert complexes. In another recent article
[arXiv:1005.4455], we extended the Arnold-Falk-Winther framework by analyzing
variational crimes (a la Strang) on Hilbert complexes. In particular, this gave
a treatment of finite element exterior calculus on manifolds, generalizing
techniques from surface finite element methods and recovering earlier a priori
estimates for the Laplace-Beltrami operator on 2- and 3-surfaces, due to Dziuk
[Lecture Notes in Math., vol. 1357 (1988), 142-155] and later Demlow [SIAM J.
Numer. Anal., 47 (2009), 805-827], as special cases. In the present article, we
extend the Hilbert complex framework in a second distinct direction: to the
study of semilinear mixed problems. We do this, first, by introducing an
operator-theoretic reformulation of the linear mixed problem, so that the
semilinear problem can be expressed as an abstract Hammerstein equation. This
allows us to obtain, for semilinear problems, a priori solution estimates and
error estimates that reduce to the Arnold-Falk-Winther results in the linear
case. We also consider the impact of variational crimes, extending the results
of our previous article to these semilinear problems. As an immediate
application, this new framework allows for mixed finite element methods to be
applied to semilinear problems on surfaces.Comment: 22 pages; v2: major revision, particularly sharpening of error
estimates in Section