Lai, M.-J. and L. L. Schumaker, Macro-elements and stable local bases for splines on Powell-Sabin triangulations, manuscript, 1999. Home/Search Document Details and Download
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Smooth Macro-Elements Based on Clough-Tocher Triangle Splits - Alfeld, Schumaker (2000) (1 citation) (Correct)
....here, and it is not possible to enforce lower supersmoothness at the vertices of T . 16 Remark 9.4. It is of course possible to construct smooth macro elements using lower degree splines provided we work with more complicated triangle splits. For macro elements on Powell Sabin splits, see [2,12]. Remark 9.5. It is also possible to create macro elements with even fewer degrees of freedom by the process of condensation. This amounts to further restricting the spline space (usually by forcing certain cross derivatives along edges of the triangle T to be of lower degree than they naturally ....
Lai, M.-J. and L. L. Schumaker, Macro-elements and stable local bases for splines on Powell-Sabin triangulations, manuscript, 1999.
On Stable Local Bases for Bivariate Polynomial Spline Spaces - Davydov, Schumaker (1999) (1 citation) (Correct)
....with d 3r 2 are known for general triangulations. However, it is possible to construct stable bases for some values of d 3r 2 for classes of splines defined on special triangulations using macro element techniques. These include Clough Tocher and Powell Sabin refinements, for example. See [25,26]. Remark 8.6. For multiresolution applications, it is important to work with sequences of triangulations which are nested. In such cases, the corresponding spline spaces S r d (4) are also nested, but in general the various superspline subspaces are not. See the discussion of this super spline ....
Lai, M.-J. and L. L. Schumaker, Macro-elements and stable local bases for splines on Powell-Sabin triangulations, manuscript, 1999.
Macro-Elements and Stable Local Bases for Splines on.. - Lai, Schumaker (1999) (3 citations) Self-citation (Lai) (Correct)
....of macro elements. x7. Optimality of the macro elements In this section we explore to what extent the spaces chosen in (1.2) are optimal with respect to the degrees of the splines and the number of degrees of freedom of the corresponding macro elements. Fix the smoothness r. By Theorem 10.1 of [11], a necessary condition for constructing a macro element on a Clough Tocher cell is that we use splines with super smoothness ae i l 3r Gamma 1 2 m = 3m; if r = 2m, 3m 1; if r = 2m 1 (7:1) at each vertex v i of T . This means that in order to construct a macro element on the ....
....the 3 direction mesh, the spaces S r d (4) do not possess optimal order approximation power. This means that neither they (nor any subspace S containing P d ) has a stable local basis. Remark 8.4. Macro elements and stable local bases can be constructed for several other refinement methods. In [11] we do this for the well known Powell Sabin split. Remark 8.5. We would like to thank Peter Alfeld for writing a beautiful JAVA program which computes dimensions of spline spaces (in exact arithmetic) and which can also be used for verifying minimal determining sets. We have made extensive use of ....
Lai, M.-J. and L. L. Schumaker, Macro-elements and stable local bases for splines on Powell-Sabin triangulations, manuscript, 1999.
Macro-Elements and Stable Local Bases for Splines on.. - Lai, Schumaker (1999) (3 citations) Self-citation (Lai) (Correct)
....of macro elements. x7. Optimality of the macro elements In this section we explore to what extent the spaces chosen in (1.2) are optimal with respect to the degrees of the splines and the number of degrees of freedom of the corresponding macro elements. Fix the smoothness r. By Theorem 10.1 of [11], a necessary condition for constructing a macro element on a Clough Tocher cell is that we use splines with super smoothness ae i l 3r Gamma 1 2 m = 3m; if r = 2m, 3m 1; if r = 2m 1 (7:1) at each vertex v i of T . This means that in order to construct a macro element on the ....
....the 3 direction mesh, the spaces S r d (4) do not possess optimal order approximation power. This means that neither they (nor any subspace S containing P d ) has a stable local basis. Remark 8.4. Macro elements and stable local bases can be constructed for several other refinement methods. In [11] we do this for the well known Powell Sabin split. Remark 8.5. We would like to thank Peter Alfeld for writing a beautiful JAVA program which computes dimensions of spline spaces (in exact arithmetic) and which can also be used for verifying minimal determining sets. We have made extensive use of ....
Lai, M.-J. and L. L. Schumaker, Macro-elements and stable local bases for splines on Powell-Sabin triangulations, manuscript, 1999.
Bounds on Projections onto Bivariate Polynomial Spline.. - von Golitschek.. (2002) Self-citation (Schumaker) (Correct)
....follows from (4.1) It follows immediately from this lemma that if S is a space of splines defined on a fi quasi uniform triangulation, then the constant C 2 in (3.3) depends only on K 2 =K 1 ; K 4 =K 3 and fi; x5. Spline spaces with stable local bases The following definition is taken from [7,15,16]. Definition 5.1. We say that a basis B : fOE g 2M for a space S of splines on a triangulation 4 is a stable local basis provided 1) there exists an integer such that for each 2 M, supp(OE ) star (v ) for some vertex v of 4; 5 2) there exist constants 0 K 5 K 6 1, ....
....of neighboring vertices v; u 2 V, where k v : maxfae v ; g; v 2 V; with : r Xi r 1 2 Pi . In this case we can also take = 3. 4) Spline spaces of the form S r d(r) 4PS ) for some special values of d(r) where 4PS is the Powell Sabin refinement of an arbitrary triangulation 4, see [15]. In this case we can take = 1. 5) Spline spaces of the form S r d(r) 4CT ) for some special values of d(r) where 4CT is the Clough Tocher refinement of an arbitrary triangulation 4, see [16] Again we can take = 1. 6) Certain other superspline spaces with d 3r 2 described in [4] where ....
Lai, M. J. and L. L. Schumaker, Macro-elements and stable local bases for splines on Powell-Sabin triangulations, submitted, 1999.
Macro-Elements and Stable Local Bases for Splines on .. - Ming-Jun Lai, Larry .. (1999) (3 citations) Self-citation (Lai) (Correct)
....of macro elements. x7. Optimality of the macro elements In this section we explore to what extent the spaces chosen in (1.2) are optimal with respect to the degrees of the splines and the number of degrees of freedom of the corresponding macro elements. Fix the smoothness r. By Theorem 10.1 of [11], a necessary condition for constructing a macro element on a Clough Tocher cell is that we use splines with super smoothness ae i l 3r Gamma 1 2 m = 3m; if r = 2m, 3m 1; if r = 2m 1 (7:1) at each vertex v i of T . This means that in order to construct a macro element on the ....
....S r d (4) do not possess optimal order approximation order for arbitrary triangulations. This means that neither they (nor any subspace S containing P d ) has a stable local basis. Remark 8.4. Macro elements and stable local bases can be constructed for several other refinement methods. In [11] we do this for the well known Powell Sabin split. Remark 8.5. We would like to thank Peter Alfeld for writing a beautiful JAVA program which computes dimensions of spline spaces (in exact arithmetic) and which can also be used for verifying minimal determining sets. We have made extensive use of ....
Lai, M.-J. and L. L. Schumaker, Macro-elements and stable local bases for splines on Powell-Sabin triangulations, manuscript, 1999.
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