## Wavelet and Multiscale Methods for Operator Equations (1997)

Venue: | Acta Numerica |

Citations: | 191 - 39 self |

### BibTeX

@ARTICLE{Dahmen97waveletand,

author = {Wolfgang Dahmen},

title = {Wavelet and Multiscale Methods for Operator Equations},

journal = {Acta Numerica},

year = {1997},

volume = {6},

pages = {55--228}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of these requirements as well as for their realization. This is also particularly important for understanding the severe obstructions, that keep us at present from readily materializing all the principally promising perspectives.

### Citations

1892 |
Sobolev Spaces
- Adams
- 1975
(Show Context)
Citation Context ...ammak j ; k = 0; : : : 2 j \Gamma 1; of the box function OE(x) = ( 1; 0sxs1; 0; else; (1.2.1) form an orthonormal basis of their linear span S j relative to the standard inner product hf; gi = hf; gi =-=[0;1]-=- := 1 R 0 f(x)g(x) dx. Since OE(x) = OE(2x) + OE(2x \Gamma 1); so that OE j;k = 1 p 2 (OE j+1;2k + OE j+1;2k+1 ); (1.2.2) the S j are nested and the closure of their union relative to k \Delta k L 2 (... |

1767 | Biorthogonal bases of compactly supported wavelets
- Cohen, A, et al.
- 1992
(Show Context)
Citation Context ...an extensive discussion of this background see [80]. However, orthonormality will merely be viewed as a special case of the more flexible concept of biorthogonality that came up in Section 3.4.2; see =-=[45]-=-. 4.2.2 Dual Pairs The scaling functions OE; ~ OE are said to form a dual pair if hOE; ~ OE(\Delta \Gamma k)i IR := Z IR OE(x) ~ OE(x \Gamma k) dx = ffi 0;k ; k 2 ZZ: (4.2.12) We will sometimes refer ... |

1165 |
Mixed and Hybrid Finite Element Methods
- Brezzi, Fortin
- 1991
(Show Context)
Citation Context ...ined by hAu; vi\Omega = a(u; v); v 2 V; b(v; ) = hBv; i\Omega ;s2 M: It is well known that in both cases L is an isomorphism from H 1 := V \Theta M onto H 2 := V \Theta M , that is, (1.5.5) is valid (=-=[26, 35, 96]-=-), which in this case means that inf 2M sup v2V b(v; ) kvk V kkMsfi ? 0: (2.2.12) Note that in the case (2.2.10) the Galerkin approximation of A is a mass matrix. Introducing suitable weighted inner p... |

993 |
Raviart Finite Element Methods for Navier-Stokes Equations
- Girault
- 1986
(Show Context)
Citation Context ...ined by hAu; vi\Omega = a(u; v); v 2 V; b(v; ) = hBv; i\Omega ;s2 M: It is well known that in both cases L is an isomorphism from H 1 := V \Theta M onto H 2 := V \Theta M , that is, (1.5.5) is valid (=-=[26, 35, 96]-=-), which in this case means that inf 2M sup v2V b(v; ) kvk V kkMsfi ? 0: (2.2.12) Note that in the case (2.2.10) the Galerkin approximation of A is a mass matrix. Introducing suitable weighted inner p... |

853 |
Multigrid Methods and Applications
- Hackbusch
- 1985
(Show Context)
Citation Context ...s. Accordingly, the most successful numerical schemes exploit in one way or another the interaction of different scales of discretization. A very prominent representative is the multigrid methodology =-=[105, 27]-=-. In a way it has caused a breakthrough in numerical analysis since, in an important range of cases it does indeed provide asymptotically optimal schemes. For closely related multilevel techniques and... |

794 | A fast algorithm for particle simulations
- Greengard, Rohklin
- 1987
(Show Context)
Citation Context ...xity to almost linear growth, if the analytical background of the problem is properly exploited. Examples of this type are panel clustering [108, 109, 159] or the closely related multipole expansions =-=[128, 100, 158]-=-. A similar finite difference-based approach is presented in [34, 33]. Yet another direction has been initiated by the startling paper [18]. As announced in Section 1.5 (III), the representation of ce... |

506 |
Multiresolution approximation and wavelet orthonormal bases of L2(R d
- Mallat
- 1989
(Show Context)
Citation Context ... later in connection with preconditioning. 4.2 Wavelets on IR n The construction of wavelet bases is best understood for H = L 2 (IR), where the notion of multiresolution analysis has originated from =-=[138, 139, 79]-=-. 4.2.1 Stationary Multiresolution Let us first consider the univariate case n = 1. Suppose that OE 2 L 2 (IR) has stable shifts kck ` 2 (ZZ)sfl fl fl fl fl fl X k2ZZ c k OE(\Delta \Gamma k) fl fl fl ... |

484 |
Interpolation theory, function spaces, differential operators
- Triebel
- 1995
(Show Context)
Citation Context ...0 one can define kfkH s(\Omega\Gamma := inffkgk H s (IR n ) : g j\Omega = fg. Alternatively, H s(\Omega\Gamma can be defined by interpolation between L 2(\Omega\Gamma and H m(\Omega\Gamma1 m ? s, see =-=[13, 85, 173]-=-. When s ! 0 one can use duality. For any normed linear space V , the dual space, consisting of all 16 bounded linear functionals on V , is denoted by V . It is a Banach space under the norm kwk Vs:= ... |

470 |
The lifting scheme: a custom-design construction of biorthogonal wavelets
- Sweldens
- 1996
(Show Context)
Citation Context ...1 , varying L j and K j produces a whole family of further stable completions and corresponding decompositions of the spaces S(\Phi j ). The special case K j = I covers the lifting scheme proposed in =-=[168, 169]-=-. In this case one has \Psi T j = \Phi T j+1 M j;1 = \Phi T j+1 M j;0 L j + \Phi T j+1 M j;1 = \Phi T j L+ \Psi j ; that is, in terms of individual functions, one has / j;k = X l2\Delta j (L j ) l;k O... |

443 |
Fast wavelet transforms and numerical algorithms
- Beylkin, Coifman, et al.
- 1992
(Show Context)
Citation Context ...rator T . In particular, in the context of periodic problems the following alternate representation has been propagated by several researches. It is called non--standard (NS) form; see, for instance, =-=[18, 21, 87]-=-. While hT \Psi; \Psii T arises from the formal expansion (see (5.3.7)) T = \Sigma 0 T \Sigma 0 = 1 X j;l=0 (Q j \Gamma Q j \Gamma1 )T (Q l \Gamma Q l\Gamma1 ); (7.7.13) setting Q \Gamma1 = 0, the alt... |

434 |
A review of a posteriori error estimation and adaptive mesh-refinement techniques
- Verfürth
- 1996
(Show Context)
Citation Context ...cussed before. On the other hand, adaptive techniques have been extensively studied in the context of finite element discretizations of (primarily) elliptic differential equations; see, for instance, =-=[8, 9, 11, 24, 91, 178]-=-. These methods are based on a-posteriori error indicators or estimators. In practice they have been proven to be quite successful. However, the analysis and the schemes are rather dependent on the pa... |

427 | The lifting scheme: A construction of second generation wavelets
- Sweldens
- 1997
(Show Context)
Citation Context ...1 , varying L j and K j produces a whole family of further stable completions and corresponding decompositions of the spaces S(\Phi j ). The special case K j = I covers the lifting scheme proposed in =-=[168, 169]-=-. In this case one has \Psi T j = \Phi T j+1 M j;1 = \Phi T j+1 M j;0 L j + \Phi T j+1 M j;1 = \Phi T j L+ \Psi j ; that is, in terms of individual functions, one has / j;k = X l2\Delta j (L j ) l;k O... |

373 |
Iterative methods by space decomposition and subspace correction
- Xu
- 1992
(Show Context)
Citation Context ...does indeed provide asymptotically optimal schemes. For closely related multilevel techniques and a unified treatment of several variants as multiplicative or additive subspace correction methods see =-=[31, 146, 184, 187]-=-. Although there remain still many unresolved problems, multigrid or multilevel schemes in the classical framework of finite difference and finite element discretizations exhibit by now a comparativel... |

273 |
Parallel multilevel preconditioners
- Bramble, Pasciak, et al.
- 1990
(Show Context)
Citation Context ...does indeed provide asymptotically optimal schemes. For closely related multilevel techniques and a unified treatment of several variants as multiplicative or additive subspace correction methods see =-=[31, 146, 184, 187]-=-. Although there remain still many unresolved problems, multigrid or multilevel schemes in the classical framework of finite difference and finite element discretizations exhibit by now a comparativel... |

261 |
Wavelets on the interval and fast wavelet transforms. Applied and Computational Harmonic Analysis
- Cohen, Daubechies, et al.
- 1993
(Show Context)
Citation Context ...ages of cubes. This includes closed surfaces arising in connection with boundary integral equations (see Section 2.2). Wavelets on the interval have been discussed in several papers; see for instance =-=[5, 46, 41, 68]-=-. The basic idea common to all these approaches is to construct multiresolution sequences S on [0; 1], which up to local boundary effects, agree with the restriction of the stationary spaces defined o... |

240 | Spherical wavelets: Efficiently representing functions on the sphere
- Schröder, Sweldens
- 1995
(Show Context)
Citation Context ...-called lifting scheme [168, 169] is very similar in spirit. Its applications, for instance in Computer Graphics, also demonstrate its versatility and efficiency in connection with unstructured grids =-=[161]-=-. Unfortunately, the understanding of analytical properties like stability and norm equivalences in such a more general setting appears still to be in its infancy. Attempts to develop stability criter... |

239 | On the multi-level splitting of finite element spaces for indefinite elliptic boundary value problems
- Yserentant
- 1986
(Show Context)
Citation Context ...re nested and, of course, their union is dense in L 2 ([0; 1]). 8 In order to successively update solutions from coarser grids we consider the following hierarchical decomposition of the trial spaces =-=[186]-=-. Instead of using orthogonal projections as in Section 1.2, we consider the Lagrange projectors L j f := 2 j X k=0 2 \Gammaj =2 f(2 \Gammaj k)OE j;k ; (1.4.9) and note that the complements W j := (L ... |

215 |
Rapid solution of integral equations of classical potential theory
- Rokhlin
- 1985
(Show Context)
Citation Context ...xity to almost linear growth, if the analytical background of the problem is properly exploited. Examples of this type are panel clustering [108, 109, 159] or the closely related multipole expansions =-=[128, 100, 158]-=-. A similar finite difference-based approach is presented in [34, 33]. Yet another direction has been initiated by the startling paper [18]. As announced in Section 1.5 (III), the representation of ce... |

195 |
Error estimates for adaptive finite element computations
- Babuska, Rheinboldt
- 1978
(Show Context)
Citation Context ...cussed before. On the other hand, adaptive techniques have been extensively studied in the context of finite element discretizations of (primarily) elliptic differential equations; see, for instance, =-=[8, 9, 11, 24, 91, 178]-=-. These methods are based on a-posteriori error indicators or estimators. In practice they have been proven to be quite successful. However, the analysis and the schemes are rather dependent on the pa... |

179 |
Multilevel finite element approximation: theory and applications
- Oswald
- 1993
(Show Context)
Citation Context ...does indeed provide asymptotically optimal schemes. For closely related multilevel techniques and a unified treatment of several variants as multiplicative or additive subspace correction methods see =-=[31, 146, 184, 187]-=-. Although there remain still many unresolved problems, multigrid or multilevel schemes in the classical framework of finite difference and finite element discretizations exhibit by now a comparativel... |

175 | Some a posteriori error estimates for elliptic partial differential equations
- Bank, Weiser
- 1985
(Show Context)
Citation Context ...dressed. 1) If the boundary @\Omega is rather regular, a standard multilevel finite element scheme, at least in the 2D case, combined with the existing adaptive refinement schemes (see, for instance, =-=[24, 11]-=-) applied directly to the problem on\Omega would realize at least the same favorable complexity. 2) If the boundary has very little regularity, it is not clear how to properly balance the regularity o... |

171 | Using the refinement equation for the construction of pre-wavelets II: power of two
- Jia, Miccelli
- 1991
(Show Context)
Citation Context ...s \Phi j of this type to [0; 1]. The function OE is often called scaling function or generator of the multiresolution sequence S = fS(\Phi j )g j2ZZ , which is known to be dense in L 2 (IR); see e.g. =-=[23, 114]-=-. Time-frequency analysis and Fourier techniques have been an indispensible source of construction tools. It is well known [23, 80, 138, 114], that in terms of the Fourier transform, stability (4.2.1)... |

165 |
Compression of wavelet decompositions
- DeVore, Jawerth, et al.
- 1992
(Show Context)
Citation Context ...ngsj;k as an additional factor in (4.2.36). 5 Norm Equivalences and Function Spaces One of the most important properties of wavelets is that they can be used to characterize function spaces (see e.g. =-=[81, 139]-=-). The Riesz basis property (3.4.13) which came up in connection with the stability of multiscale transformations (Theorem 3.3) is a special case in a whole scale of similar relations. This will be se... |

164 | Introduction to adaptive methods for differential equations
- Eriksson, Estep, et al.
- 1995
(Show Context)
Citation Context ...cussed before. On the other hand, adaptive techniques have been extensively studied in the context of finite element discretizations of (primarily) elliptic differential equations; see, for instance, =-=[8, 9, 11, 24, 91, 178]-=-. These methods are based on a-posteriori error indicators or estimators. In practice they have been proven to be quite successful. However, the analysis and the schemes are rather dependent on the pa... |

156 | On the representation of operators in bases of compactly supported wavelets - Beylkin - 1992 |

139 |
On the fast matrix multiplication in the boundary element method by panel clustering
- Hackbusch, Nowak
- 1989
(Show Context)
Citation Context ...es this indeed allows one to reduce the computational complexity to almost linear growth, if the analytical background of the problem is properly exploited. Examples of this type are panel clustering =-=[108, 109, 159]-=- or the closely related multipole expansions [128, 100, 158]. A similar finite difference-based approach is presented in [34, 33]. Yet another direction has been initiated by the startling paper [18].... |

126 |
New Thoughts on Besov Spaces
- Peetre
- 1976
(Show Context)
Citation Context ...f; 2 \Gammaj ) 2 ; (5.2.16) which closes the gap. These results are closely related to interpolation theory. In fact, the A s Q are interpolation spaces obtained by the real method ; see for instance =-=[13, 85, 84, 150]-=-. A detailed discussion of this point of view can be found in [61]. As mentioned before, the role of !(\Delta; t) is typically played by a K-functional or a modulus of smoothness, which under our assu... |

123 |
Non-separable bidimensional wavelet Bases, Rev
- Cohen, Daubechies
- 1993
(Show Context)
Citation Context ...er, to reduce the number of mother wavelets, one might employ scalings by suitable integer matrices M with all eigenvalues strictly greater than one. One then needs j det M j \Gamma 1 mother wavelets =-=[104, 47, 62]-=-. Again, much less machinery is available in this case. Finally, instead of considering spaces generated by a single scaling function, one can use a fixed finite collection of generators. In summary, ... |

122 |
Refinement Algorithms and Data Structures for Regular Local Mesh Refinement
- Bank, Sherman, et al.
- 1983
(Show Context)
Citation Context ...ing stable bases for the resulting finite element spaces. If one wants to avoid slave nodes, the nonuniform refinements have to be closed by introducing suitable transition elements; see for instance =-=[10]-=-. In this case, the submatrix argument does not work in a strict sense. Nevertheless, one can prove that for such adaptive refinements resulting in highly nonuniform meshes, the BPX scheme still produ... |

122 |
A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems
- Bramble, Pasciak
- 1988
(Show Context)
Citation Context ...1.8) also follows in this case from Proposition 7.9. Solving the Linear Systems Since the matrix in (7.11.7) is indefinite, the solution of (7.11.7) by iterative methods requires a bit more care; see =-=[28]-=-. The upshot of all the options is that, whenever a good preconditioner for the (positive definite) matrix A j;ff , as well as for the Schur complement K j;ff := B j A \Gamma1 j;ff B T j , is availabl... |

119 |
A convergent adaptive algorithm for Poisson’s equation
- Dörfler
- 1996
(Show Context)
Citation Context ...stimates of the above type were first obtained in [15]. Under much more specialized assumptions, results of similar nature have also been established in the finite element context; see, for instance, =-=[86]-=-. 11.4 Convergence of an Adaptive Refinement Scheme In the present setting, it can be shown with the aid of Theorem 11.1 that, under mild assumptions on the right hand-side f , a suitable adaptive cho... |

112 |
Finite Elements. Theory, Fast Solvers and
- Braess
- 2001
(Show Context)
Citation Context ...ined by hAu; vi\Omega = a(u; v); v 2 V; b(v; ) = hBv; i\Omega ;s2 M: It is well known that in both cases L is an isomorphism from H 1 := V \Theta M onto H 2 := V \Theta M , that is, (1.5.5) is valid (=-=[26, 35, 96]-=-), which in this case means that inf 2M sup v2V b(v; ) kvk V kkMsfi ? 0: (2.2.12) Note that in the case (2.2.10) the Galerkin approximation of A is a mass matrix. Introducing suitable weighted inner p... |

108 |
Interpolation of Besov spaces
- DeVore, Popov
- 1988
(Show Context)
Citation Context ...0 one can define kfkH s(\Omega\Gamma := inffkgk H s (IR n ) : g j\Omega = fg. Alternatively, H s(\Omega\Gamma can be defined by interpolation between L 2(\Omega\Gamma and H m(\Omega\Gamma1 m ? s, see =-=[13, 85, 173]-=-. When s ! 0 one can use duality. For any normed linear space V , the dual space, consisting of all 16 bounded linear functionals on V , is denoted by V . It is a Banach space under the norm kwk Vs:= ... |

104 | Multilevel preconditioning
- Dahmen, Kunoth
- 1992
(Show Context)
Citation Context ...d for t 6= 0 and large #. However, a diagonal symmetric scaling suffices to remedy this. This observation has been made on various different levels of generality in several papers (see, for instance, =-=[20, 63, 67, 72, 113, 147]-=-). 54 Theorem 6.1 ([72]) Suppose that the Galerkin scheme (6.1.6) is stable (6.1.7) and that the parameters fl; ~ fl in (6.1.1) satisfy jtj ! fl; ~ fl: (6.2.2) Let D ssbe the diagonal matrix defined b... |

93 |
Pseudo-differential Operators
- Kumano-go
- 1974
(Show Context)
Citation Context ...ng lines, which work for a much wider class of pseudo-differential operators. In fact, for smooth \Gamma these operators are classical pseudo-differential operators characterized by their symbol, see =-=[110, 123]-=-. Equation (1.5.5) follows from the boundedness of L, its injectivity on H 1 , and coercivity of the principal part of its symbol. The advantages of the approach are obvious. A 3D discretization of a ... |

92 | Stability of multiscale transformations
- Dahmen
- 1996
(Show Context)
Citation Context ...is permits the interaction of different scales of discretizations. In basis or transform oriented methods this is effected with the aid of appropriate multiscale bases of hierarchical type. Following =-=[37, 59, 60, 61]-=- a general framework of multiresolution and multiscale decompositions of trial spaces is described next in a form which will later host all the required specializations. The examples in Sections 1.2 a... |

89 | Biorthogonal spline-wavelets on the interval - Stability and moment conditions - Dahmen, Kunoth, et al. - 1999 |

86 | On the construction of multivariate (pre–) wavelets
- Boor, DeVore, et al.
- 1993
(Show Context)
Citation Context ...s \Phi j of this type to [0; 1]. The function OE is often called scaling function or generator of the multiresolution sequence S = fS(\Phi j )g j2ZZ , which is known to be dense in L 2 (IR); see e.g. =-=[23, 114]-=-. Time-frequency analysis and Fourier techniques have been an indispensible source of construction tools. It is well known [23, 80, 138, 114], that in terms of the Fourier transform, stability (4.2.1)... |

84 |
Wavelet-like bases for the fast solution of second-kind integral equations
- Alpert, Beylkin, et al.
- 1993
(Show Context)
Citation Context ...r, there is, of course, the restriction s ! 1=2, which will be seen later to be an unfortunate obstruction. 9.3 Multi--Wavelets The above geometric setting suggests the following natural concept; see =-=[2, 3, 153, 154]-=-. Let \Pi d be the set of polynomials of total degree less than d on IR n and let P := fPs: jj =s1 + : : : +sn ! dg be an orthonormal basis of \Pi d on 2, which can be generated by the Gram--Schmidt p... |

84 | Wavelets on manifolds I: Construction and domain decomposition
- Dahmen, Schneider
- 1999
(Show Context)
Citation Context ...nding univariate generator and wavelet functions. When OE is the standard piecewise linear tent function (1.4.4) and the dual bases are exact of order 2 as well, the mask coefficients can be found in =-=[68, 76]-=-). In most cases, however, so called piecewise linear pre-wavelets have been used; see, for instance, [103]. Interior and boundary wavelets are shown below in Figure 6.6.3. Figure 6.6.3: Pre-wavelets ... |

79 |
Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur effizienten Lösung grosser vollbesetzter Gleichungssysteme
- Schneider
- 1998
(Show Context)
Citation Context ...ns. This includes the Lam'e Navier equations of linearized, three-dimensional elasticity, [182], the oblique derivative problem [142], arising in physical geodesy, the exterior Stokes flow [126], see =-=[160]-=- for a brief overview. Here it suffices to describe a simple example that exhibits the principal features of this class of problem. Consider the boundary value problem \DeltaU = 0; on\Omega ; @sU = f;... |

74 | Wavelets on closed subsets on the real line
- Anderson, Hall, et al.
- 1993
(Show Context)
Citation Context ...ages of cubes. This includes closed surfaces arising in connection with boundary integral equations (see Section 2.2). Wavelets on the interval have been discussed in several papers; see for instance =-=[5, 46, 41, 68]-=-. The basic idea common to all these approaches is to construct multiresolution sequences S on [0; 1], which up to local boundary effects, agree with the restriction of the stationary spaces defined o... |

73 |
Wavelet methods for fast resolution of elliptic problems
- Jaffard
- 1992
(Show Context)
Citation Context ...d for t 6= 0 and large #. However, a diagonal symmetric scaling suffices to remedy this. This observation has been made on various different levels of generality in several papers (see, for instance, =-=[20, 63, 67, 72, 113, 147]-=-). 54 Theorem 6.1 ([72]) Suppose that the Galerkin scheme (6.1.6) is stable (6.1.7) and that the parameters fl; ~ fl in (6.1.1) satisfy jtj ! fl; ~ fl: (6.2.2) Let D ssbe the diagonal matrix defined b... |

67 |
The American Community
- Cohen, Brawer
- 2003
(Show Context)
Citation Context ...d above. Thus the biorthogonal bases constructed in Sections 4.2 and 4.4 are indeed Riesz bases. A few comments on the proof of Theorem 5.6 are in order; see [60] for details. First one observes that =-=[44]-=- NQ (\Delta) ! k \Delta kH holds if and only if k \Delta kH ! NQs(\Delta): Thus it suffices to prove that k \Delta kH ! NQ (\Delta) and k \Delta kH ! NQs(\Delta); (5.2.8) or the corresponding pair of ... |

67 | Stable Multiscale Bases and Local Error Estimation for Elliptic Problems
- Dahlke, Dahmen, et al.
(Show Context)
Citation Context ...oal: keep the computational work proportional to the number of significant terms in the wavelet expansion of the searched object, which in some sense should reflect its intrinsic complexity, see e.g. =-=[21, 54]-=-. The potential of this point of view will be one of the main themes of subsequent discussions. Wavelets are in some sense much more sophisticated tools than conventional discretizations. It will be s... |

67 |
On the abstract theory of additive and multiplicative Schwarz algorithms
- Griebel, Oswald
- 1995
(Show Context)
Citation Context ...incorporating domain decomposition and multigrid techniques. For a more extensive treatment of these issues, as well as further details concerning the following discussion, we refer, for instance, to =-=[102, 146, 184, 187]-=-. As above, L will be selfadjoint positive definite on some separable Hilbert space H = H 1 , that is, a(u; v) := hLu; vi is a symmetric bilinear form and we assume that (6.1.9) holds with H t replace... |

64 |
Signal processing and compression with wavelet packets
- Coifman, Meyer, et al.
- 1992
(Show Context)
Citation Context ...3), which has been proposed, for instance, by [116]. It aims at realizing best possible compressionsof the approximate solution by employing the concept of wavelet packets and best bases developed by =-=[49, 50]-=-. This technique is also used by [89]. Therefore we will briefly indicate some of the ideas in [116], where further details and relevant references can be found. To describe the concept of wavelet pac... |

64 |
Multiscale methods for pseudodifferential equations
- Dahmen, Prossdorf, et al.
- 1996
(Show Context)
Citation Context ...plication to collocation. When L is a pseudo-differential operator, its injectivity, boundedness and coercivity of the principal part of its symbol also imply stability (6.1.7) of the Galerkin scheme =-=[73, 72, 110]-=-. Of course, when L is selfadjoint in the sense that a(u; v) := hLu; vi (6.1.8) 53 is a symmetric bilinear form, ellipticity (2.3.1) means that k \Delta k 2 := a(\Delta; \Delta) 2sk \Delta k H t ; (6.... |

64 |
Multiresolution algorithms for the numerical solution of hyperbolic conservation laws
- Harten
- 1995
(Show Context)
Citation Context ...ations, which in a multigrid context recover the usual multigrid efficiency for elliptic problems even in the case of strong convection terms. Discrete multiresolution concepts have been developed in =-=[111]-=-, with special emphasis on the treatment of hyperbolic conservation laws. It is well-known that such systems can be viewed as evolution equations for cell averages. This fact serves as the basis for f... |

61 |
A basic norm equivalence for the theory of multilevel methods
- Bornemann, Yserentant
- 1993
(Show Context)
Citation Context ...e refinements resulting in highly nonuniform meshes, the BPX scheme still produces uniformly bounded condition numbers. This has been shown first in [63], where further details can be found; see also =-=[25]-=-. An analogous result holds for forth order problems. As a model case, one could considersL = \Delta 2 with homogeneous Dirichlet boundary conditions. A convenient conforming finite element discretiza... |