8 research outputs found
Standard finite elements for the numerical resolution of the elliptic Monge-Ampere equation: Aleksandrov solutions
We prove a convergence result for a natural discretization of the Dirichlet
problem of the elliptic Monge-Ampere equation using finite dimensional spaces
of piecewise polynomial C0 or C1 functions. Standard discretizations of the
type considered in this paper have been previous analyzed in the case the
equation has a smooth solution and numerous numerical evidence of convergence
were given in the case of non smooth solutions. Our convergence result is valid
for non smooth solutions, is given in the setting of Aleksandrov solutions, and
consists in discretizing the equation in a subdomain with the boundary data
used as an approximation of the solution in the remaining part of the domain.
Our result gives a theoretical validation for the use of a non monotone finite
element method for the Monge-Amp\`ere equation
Eliminating Gibbs Phenomena: A Non-linear Petrov-Galerkin Method for the Convection-Diffusion-Reaction Equation
In this article we consider the numerical approximation of the
convection-diffusion-reaction equation. One of the main challenges of designing
a numerical method for this problem is that boundary layers occurring in the
convection-dominated case can lead to non-physical oscillations in the
numerical approximation, often referred to as Gibbs phenomena. The idea of this
article is to consider the approximation problem as a residual minimization in
dual norms in Lq-type Sobolev spaces, with 1 < q < . We then apply a
non-standard, non-linear PetrovGalerkin discretization, that is applicable to
reflexive Banach spaces such that the space itself and its dual are strictly
convex. Similar to discontinuous Petrov-Galerkin methods, this method is based
on minimizing the residual in a dual norm. Replacing the intractable dual norm
by a suitable discrete dual norm gives rise to a non-linear inexact mixed
method. This generalizes the Petrov-Galerkin framework developed in the context
of discontinuous Petrov-Galerkin methods to more general Banach spaces. For the
convection-diffusion-reaction equation, this yields a generalization of a
similar approach from the L2-setting to the Lq-setting. A key advantage of
considering a more general Banach space setting is that, in certain cases, the
oscillations in the numerical approximation vanish as q tends to 1, as we will
demonstrate using a few simple numerical examples
Gibbs Phenomena for -Best Approximation in Finite Element Spaces -- Some Examples
Recent developments in the context of minimum residual finite element methods
are paving the way for designing finite element methods in non-standard
function spaces. This, in particular, permits the selection of a solution space
in which the best approximation of the solution has desirable properties. One
of the biggest challenges in designing finite element methods are non-physical
oscillations near thin layers and jump discontinuities. In this article we
investigate Gibbs phenomena in the context of -best approximation of
discontinuities in finite element spaces with . Using carefully
selected examples, we show that on certain meshes the Gibbs phenomenon can be
eliminated in the limit as tends to . The aim here is to show the
potential of as a solution space in connection with suitably designed
meshes
Eliminating Gibbs phenomena: A non-linear Petrov–Galerkin method for the convection–diffusion–reaction equation
In this article we consider the numerical approximation of the convection-diffusion-reaction equation. One of the main challenges of designing a numerical method for this problem is that boundary layers occurring in the convection-dominated case can lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The idea of this article is to consider the approximation problem as a residual minimization in dual norms in Lq-type Sobolev spaces, with 1 < q < . We then apply a non-standard, non-linear PetrovGalerkin discretization, that is applicable to reflexive Banach spaces such that the space itself and its dual are strictly convex. Similar to discontinuous Petrov-Galerkin methods, this method is based on minimizing the residual in a dual norm. Replacing the intractable dual norm by a suitable discrete dual norm gives rise to a non-linear inexact mixed method. This generalizes the Petrov-Galerkin framework developed in the context of discontinuous Petrov-Galerkin methods to more general Banach spaces. For the convection-diffusion-reaction equation, this yields a generalization of a similar approach from the L2-setting to the Lq-setting. A key advantage of considering a more general Banach space setting is that, in certain cases, the oscillations in the numerical approximation vanish as q tends to 1, as we will demonstrate using a few simple numerical examples
Gibbs phenomena for Lq-best approximation in finite element spaces
Recent developments in the context of minimum residual finite element methods are paving the way for designing quasi-optimal discretization methods in non-standard function spaces, such as L q-type Sobolev spaces. For q → 1, these methods have demonstrated huge potential in avoiding the notorious Gibbs phenomena, i.e., the occurrence of spurious non-physical oscillations near thin layers and jump discontinuities. In this work we provide theoretical results that explain some of these numerical observations. In particular, we investigate the Gibbs phenomena for L q-best approximations of discontinuities in finite element spaces with 1 ≤ q < ∞. We prove sufficient conditions on meshes in one and two dimensions such that over-and undershoots vanish in the limit q → 1. Moreover, we include examples of meshes such that Gibbs phenomena remain present even for q = 1 and demonstrate that our results can be used to design meshes so as to eliminate the Gibbs phenomenon
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Wavelet and Multiscale Methods
Various scientific models demand finer and finer resolutions of relevant features. Paradoxically, increasing computational power serves to even heighten this demand. Namely, the wealth of available data itself becomes a major obstruction. Extracting essential information from complex structures and developing rigorous models to quantify the quality of information leads to tasks that are not tractable by standard numerical techniques. The last decade has seen the emergence of several new computational methodologies to address this situation. Their common features are the nonlinearity of the solution methods as well as the ability of separating solution characteristics living on different length scales. Perhaps the most prominent examples lie in multigrid methods and adaptive grid solvers for partial differential equations. These have substantially advanced the frontiers of computability for certain problem classes in numerical analysis. Other highly visible examples are: regression techniques in nonparametric statistical estimation, the design of universal estimators in the context of mathematical learning theory and machine learning; the investigation of greedy algorithms in complexity theory, compression techniques and encoding in signal and image processing; the solution of global operator equations through the compression of fully populated matrices arising from boundary integral equations with the aid of multipole expansions and hierarchical matrices; attacking problems in high spatial dimensions by sparse grid or hyperbolic wavelet concepts. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computation and to promote the exchange of ideas emerging in various disciplines