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At least 271 records · Page 15

Angrite LEW87051: Are the olivines pheno's or xeno's? A continuing story

The achondrite LEW87051 is a porphyritic basalt consisting of large subhedral to euhedral zoned olivines in a finer-grained groundmass. The texture of this groundmass looks remarkably like a quenched melt. However, although the rock is clearly igneous, its exact origins and history are under dispute. From petrographic observations, Prinz felt that the large olivines were xenocrysts and that the zoning reflected interaction with an unrelated, CAI-enriched melt. McKay et al. was able to model the olivines as phenocrysts, whose zoning was the result of a parent melt that changed in composition as material crystallized, e.g., fractional crystallization in a closed system, and calculated a parent melt composition. Jurewicz and McKay compared the calculated parent melt composition with actual partial melts from CV and CM chondrites. They showed that the calculated melt was substantially different from equilibrium melts of these chondrites; however, the LEW87051 groundmass composition was similar to some of the low temperature partial melts, although slightly enriched in AN (or depleted in OL) components. This study presents the results of an independent petrologic look at other olivines in LEW87051 and the preliminary results of a quantitative model for the major zoning in these olivines as diffusive-exchange with an olivine-saturated, low temperature angritic melt.

Jurewicz, A. J. G.↗

On possible Mn-53 heterogeneity in the early solar system

The effects of influence of shock wave propagation on the energy spectrum of accelerated particles that lead to different production rates of radionuclides, in particular, Mn-53, on small scales in the early solar system are shown. Search for evidence for extinct Mn-53 has stimulated investigations of Cr isotope anomalies in meteorites. The linear correlation between the magnitude of the Cr-53* excesses and the Mn/Cr ratio that unambiguously proves the in situ decay of Mn-53 was detected, really, in different mineral phases of some carbonaceous and enstatite chondrites, primitive achondrites, pallasites and iron meteorites. However, the data on the Cr-53* excess rarely defines a single linear array on a Mn-53-Cr-52 evolution diagram even for meteorites of the same chemical group. A clear isochron with Mn-53/Mn-55 = 4.4 plus or minus 1.0 x 10(exp -5) (in range of approximately 2.4 to approximately 9 x 10(exp -5)) is observed for CAI of the Allende C3-chondrite while the data for the Murchison C2- and Orgueil C1-chondrites fall much lower corresponding rather to Mn-53/Mn-55 less than 2 x 10(exp -5). In the case of iron meteorites it ranges from less than 5 x 10(exp -8) to less than 5 x 10(exp -5).

Lavrukhina, A. K.↗

Correlated petrographic, electron microprobe, and ion microprobe studies of selected primitive and processed phase assemblages in meteorites

During the past three years we have received support to continue our research in elucidating the formation and alteration histories of selected meteoritic materials by a combination of petrographic, trace element, and isotopic analyses employing optical and scanning electron microscopes and electron and ion microprobes. The awarded research funds enabled the P.I. to attend the annual LPSC, the co-I to devote approximately 15 percent of his time to the research proposed in the grant, and partial support for a visiting summer post-doctoral fellow to conduct electron microprobe analyses of meteoritic samples in our laboratory. The research funds, along with support from the NASA Education Initiative awarded to P.I. G. Wasserburg, enabled the co-I to continue a mentoring program with inner-city minority youth. The support enabled us to achieve significant results in the five projects that we proposed (in addition to the Education Initiative), namely: studies of the accretional and post-accretional alteration and thermal histories in CV meteorites, characterization of periclase-bearing Fremdlinge in CV meteorites, characterization of Ni-Pt-Ge-Te-rich Fremdlinge in CV meteorites in an attempt to determine the constraints they place on the petrogenetic and thermal histories of their host CAI's, correlated electron and ion microprobe studies of silicate and phosphate inclusions in the Colomera meteorite in an attempt to determine the petrogenesis of the IE iron meteorites, and development of improved instrumental and correction procedures for improved accuracy of analysis of meteoritic materials with the electron microprobe. This grant supported, in part or whole, 18 publications so far by our research team, with at least three more papers anticipated. The list of these publications is included. The details of the research results are briefly summarized.

Albee, Arden L.↗

Papers presented to the Conference on Chondrules and the Protoplanetary Disk

The following topics are covered in the presented papers: (1) producing chondrules; (2) carbons, CAI's, and chondrules; (3) large scale processes in the solar nebula; (4) chondrule-matrix relationships in chondritic meteorites; (5) overview of nebula models; (6) constraints placed on the nature of chondrule precursors; (7) turbulent diffusion and concentration of chondrules in the protoplanetary nebula; (8) heating and cooling in the solar nebula; (9) crystallization trends of precursor pyroxene in ordinary chondrites; (10) precipitation induced vertical lightning in the protoplanetary nebula; (11) the role of chondrules in nebular fractionations of volatiles and other elements; (12) astronomical observations of phenomena in disks; (13) experimental constraints on models for origins of chondrules, and various other topics.

Source record↗

The origin of refractory precursor components of chondrules

A percentage of all chondrules from carbonaceous chondrites are intermediate in major-element composition between the rare Ca,Al-rich chondrules and the more abundant ferromagnesian chondrules. Some of them possess fractionated REE and trace-element patterns similar to those in CAI's (Ca-Al-rich inclusions). The abundance patterns are controlled primarily by volatilities expected from a solar composition. On the basis of refractory lithophile fractionations, it is suggested that one of the refractory precursors of C chondrules is a condensate from the nebular gas or an evaporative residue of primitive dust clumps.

Misawa, K.↗

Siderophile elements and metal-silicate fractionation in the solar nebula

The most important nebular fractionation affecting the siderophile elements is the metal-silicate fractionation process and its relationship to the chondrule formation process is poorly understood. Understanding these processes is important in terms of understanding the expected compositional range for planetary building blocks. In a general way the composition of chondrites can be derived from the composition of the CI chondrites by addition or subtraction of a refractory component similar to CAI's (in some cases with a Mg2SiO4 component) and by addition or subtraction of Fe metal. Thus normalization to Fe produces the least spread in the normalized abundances of most siderophile elements relative to CI abundances. Detailed bulk chemical studies of chondrules have shown that their siderophile elements have refractory-element siderophile-element signatures (for example, Ir/Ni) that are similar to the bulk meteorites, but distinct among the different chondrite groups. This data suggests that the chondrules were not supplied to the chondrule groups from a single homogeneous source, and that each chondrite group's characteristic siderophile-element signature was established before chondrule formation. These authors make a further inference that all siderophile-lithophile-element fractionation occurred before chondrule formation, but recent discoveries and observations suggest this is not true. The discovery of the metal-rich CH meteorites, the recognition of the role of aerodynamic forces in the solar nebula, and the possible role of metal-silicate segregation during the chondrule formation process suggests that metal-silicate fractionation occurred before, during and after chondrule formation.

Newsom, H. E.↗

Chemical production of chondrule oxygen isotopic composition

Defining the source of observed meteoritic O isotopic anomalies remains a fundamental challenge. The O isotopic composition of chondrules are particularly striking. There are at least three types of chemical processes that produce the isotopic compositions observed in chondrules and Ca-Al-rich inclusions (CAI's). The processes are rather general, viz, they require no specialized processes and the processes associated with chondrule production are likely to produce the observed compositions.

Thiemens, M. H.↗

Static and dynamic strain energy release rates in toughened thermosetting composite laminates

In this work, the static and dynamic fracture properties of several thermosetting resin based composite laminates are presented. Two classes of materials are explored. These are homogeneous, thermosetting resins and toughened, multi-phase, thermosetting resin systems. Multi-phase resin materials have shown enhancement over homogenous materials with respect to damage resistance. The development of new dynamic tests are presented for composite laminates based on Width Tapered Double Cantilevered Beam (WTDCB) for Mode 1 fracture and the End Notched Flexure (ENF) specimen. The WTDCB sample was loaded via a low inertia, pneumatic cylinder to produce rapid cross-head displacements. A high rate, piezo-electric load cell and an accelerometer were mounted on the specimen. A digital oscilloscope was used for data acquisition. Typical static and dynamic load versus displacement plots are presented. The ENF specimen was impacted in three point bending with an instrumented impact tower. Fracture initiation and propagation energies under static and dynamic conditions were determined analytically and experimentally. The test results for Mode 1 fracture are relatively insensitive to strain rate effects for the laminates tested in this study. The test results from Mode 2 fracture indicate that the toughened systems provide superior fracture initiation and higher resistance to propagation under dynamic conditions. While the static fracture properties of the homogeneous systems may be relatively high, the apparent Mode 2 dynamic critical strain energy release rate drops significantly. The results indicate that static Mode 2 fracture testing is inadequate for determining the fracture performance of composite structures subjected to conditions such as low velocity impact. A good correlation between the basic Mode 2 dynamic fracture properties and the performance is a combined material/structural Compression After Impact (CAI) test is found. These results underscore the importance of examining rate-dependent behavior for determining the longevity of structures manufactured from composite materials.

Cairns, Douglas S.↗

Assessment of pollution prevention and control technology for plating operations

The National Center for Manufacturing Sciences (NCMS) is sponsoring an on-going project to assess pollution prevention and control technology available to the plating industry and to make this information available to those who can benefit from it. Completed project activities include extensive surveys of the plating industry and vendors of technologies and an indepth literature review. The plating industry survey was performed in cooperation with the National Association of Metal Finishers. The contractor that conducted the surveys and prepared the project products was CAI Engineering. The initial products of the project were made available in April, 1994. These products include an extensive report that presents the results of the surveys and literature review and an electronic database. The project results are useful for all those associated with pollution prevention and control in the plating industry. The results show which treatment, recovery and bath maintenance technologies have been most successful for different plating processes and the costs for purchasing and operating these technologies. The project results also cover trends in chemical substitution, the identification of compliance-problem pollutants, sludge generation rates, off-site sludge recovery and disposal options, and many other pertinent topics.

Chalmer, Paul D.↗

Theoretical, observational, and isotopic estimates of the lifetime of the solar nebula

There are a variety of isotopic data for meteorites which suggest that the protostellar nebula existed and was involved in making planetary materials for some 10(exp 7) yr or more. Many cosmochemists, however, advocate alternative interpretations of such data in order to comply with a perceived constraint, from theoretical considerations, that the nebula existed only for a much shorter time, usually stated as less than or = 10(exp 6) yr. In this paper, we review evidence relevant to solar nebula duration which is available through three different disciplines: theoretical modelling of star formation, isotopic data from meteorites, and astronomical observations of T Tauri stars. Theoretical models based on observations of present star-forming regions indicate that stars like the Sun form by dynamical gravitational collapse of dense cores of cold molecular clouds in the interstellar medium. The collapse to a star and disk occurs rapidly on a time scale of the order 10(exp 5) yr. Disks evolve by dissipating energy while redistributing angular momentum, but it is difficult to predict the rate of evolution, particularly for low mass (compared to the star) disks which nonetheless still contain enough material to account for the observed planetary system. There is no compelling evidence, from available theories of disk structure and evolution, that the solar nebula must have evolved rapidly and could not have persisted for more than 1 Ma. In considering chronologically relevant isotopic data for meteorites, we focus on three methodologies: absolute ages by U-Pb/Pb-Pb, and relative ages by short-lived radionuclides (especially Al-26) and by evolution of Sr-87/Sr-86. Two kinds of meteoritic materials-refractory inclusions such as CAIs and differentiated meteorites (eucrites and angrites) - appear to have experience potentially dateable nebular events. In both case, the most straightforward interpretations of the available data indicate nebular events spanning several Ma. We also consider alternative interpretations, particularly the hypothesis of radically heterogeneous distribution of Al-26, which would avoid these chronological interpretations. The principal impetus for such alternative interpretations seems to be precisely the obviation of the chronological interpretation (i.e., the presumption rather than the inference of a short (less than or = Ma) lifetime of the nebula). Astronomical observations of T Tauri stars indicate that the presence of dusty disks is a common if not universal feature, that the disks are massive enough to accomodate a planetary system such as ours, and that at least some persist for 10(exp 7) yr or more. The results are consistent with the time scales inferred from the meteorite isotopic data. They cannot be considered conclusive with regard to solar nebula time scales, however,in part because it is difficult to relate disk observations to processes that affect meteorites, and in part because the ages assigned for these stars could be wrong by a factor of several in either direction. We conclude that the balance of available evidence favors the view that the nebula existed and was active for at least several Ma. However, because the evidence is not definitive, it is important that the issue be perceived to be an open question, whose answer should be sought rather than presumed.

Podosek, Frank A.↗

A new titanium-bearing calcium aluminosilicate phase. 2: Crystallography and crystal chemistry of grains formed in slowly cooled melts with bulk compositions of calcium-aluminium-rich inclusions

The crystallography and crystal chemistry of a new calcium- titanium-aluminosilicate mineral (UNK) observed in synthetic analogs to calcium-aluminum-rich inclusions (CAIs) from carbonaceous chondrites was studied by electron diffraction techniques. The unit cell is primitive hexagonal or trigonal, with a = 0.790 +/- 0.02 nm and c = 0.492 +/- 0.002 nm, similar to the lattice parameters of melilite and consistent with cell dimensions for crystals in a mixer furnace slag described by Barber and Agrell (1994). The phase frequently displays an epitactic relationship in which melilite acts as the host, with (0001)(sub UNK) parallel (001)(sub mel) and zone axis group 1 0 -1 0(sub UNK) parallel zone axis group 1 0 0(sub mel). If one of the two space groups determined by Barber and Agrell (1994) for their sample of UNK is applicable (P3m1 or P31m), then the structure is probably characterized by puckered sheets of octahedra and tetrahedra perpendicular to the c-axis with successive sheets coordinated by planar arrays of Ca. In this likely structure, each unit cell contains three Ca sites located in mirror planes, one octahedrally coordinated cation located along a three-fold axis and five tetrahedrally coordinated cations, three in mirrors and two along triads. The octahedron contains Ti but, because there are 1.3-1.9 cations of Ti/formula unit, some of the Ti must also be in tetrahedral coordination, an unusual but not unprecedented situation for a silicate. Tetrahedral sites in mirror planes would contain mostly Si, with lesser amounts of Al while those along the triads correspondingly contain mostly Al with subordinate Ti. The structural formula, therefore, can be expressed as Ca(sub 3)(sup VIII)(Ti,Al)(sup VI)(Al,Ti,Si)(sub 2)(sup IV)(Si,Al)(sub 3)(sup IV)O14 with Si + Ti = 4. Compositions of meteoritic and synthetic Ti-bearing samples of the phase can be described in terms of a binary solid solution between the end-members Ca3TiAl2Si3O14 and Ca3Ti(AlTi)(AlSi2)O14. A Ti-free analog with a formula of Ca3Al2Si4O14 synthesized by Paque et al. (1994) is thought to be related structurally but with the octahedral site being occupied by Al, that is, Ca(sub 3)(sup VIII)Al(sup VI)(Al,Si)(sub 2)(sup IV)(Si)(sub 3)(sup IV)O14.

Barber, David J.↗

Mn-Cr isotopic systematics of individual Chainpur chondrules

Twenty-eight chondrules separated from Chainpur (LL3.4) were surveyed for abundances of Mn, Cr, Na, Fe, Sc, Hf, Ir, and Zn by Instrumental Neutron Activation Analysis (INAA). Six, weighting 0.6-1.5 mg each, were chosen for Scanning Electron Microscopy (SEM)/Energy Dispersive X-ray (EDX) and high-precision Ce-isotopic studies. LL-chondrite-normalized (Mn/Fe)(sub LL) and (Sc/Fe)(sub LL) were found to be useful in categorizing them. Five chondrules (CH-16, -17, -18, -23, and -28) were in the range 0.5 less than (Mn/Fe)(sub LL) less than 1. 4 and 0.5 less than (Sc/Fe)(sub LL) less than 1.4. The sixth (CH-25) had (Mn/Fe)(sub LL) and (Sc/Fe)(sub LL) ratios of 0.40 and 8.1, respectively, and was enriched in the refractory lithophile elements Sc and Hf and the refractory siderophile element Ir by 2.7 and 4.4x LL abundances respectively. SEM/EDX of exterior surfaces of the chondrules showed they consisted of varying proportions of low- and high-Ca pyroxenes, olivine, glass, kamacite/taenite, and Fe-sulfides. Chromium-53/chromium-52 for the six chondrules and bulk Chainpur (WR) are presented. Chromium-54/chromium-52 is close to terrestrial and does not correlate with Mn/Cr. We provisionally ignore the possibility of initial Cr isotopic heterogeneities among the chondrules. Omitting both the CH-25 and WR data, a linear regression gives initial (Mn-53/Mn-55)(sub I) = 8 +/- 4 x 10(exp -6), corresponding to chondrule formation at Delta(t)(sub LEW) = -9 +/- 4 Ma prior to igneous crystallization of the LEW 86010 angrite. If initial (Mn-53/Mn-55)(sub 0) in the solar system were as high as approximately 4.4 x 10(exp -5) when Allende CAI formed, our data suggest Chainpur chondrules formed approximately 9 Ma later, in qualitative agreement with 'late' I-Xe formation ages for most Chainpur chondrules.

Nyquist, L.↗

Theoretical, observational, and isotopic estimates of the lifetime of the solar nebula

There are a variety of isotopic data for meteorites which suggest that the protostellar nebula existed and was involved in making planetary materials for some 10(exp 7) yr or more. Many cosmochemists, however, advocate alternative interpretations of such data in order to comply with a perceived constraint, from theoretical considerations, that the nebula existed only for a much shorter time, usually stated as less than or equal to 10(exp 6) yr. In this paper, we review evidence relevant to solar nebula duration which is available through three different disciplines: theoretical modeling of star formation, isotopic data from meteorites, and astronomical observations of T Tauri stars. Theoretical models based on observations of present star-forming regions indicate that stars like the Sun form by dynamical gravitational collapse of dense cores of cold molcular clouds in the interstellar clouds in the interstellar medium. The collapse to a star and disk occurs rapidly, on a time scale of the order 10(exp 5) yr. Disks evolve by dissipating energy while redistributing angular momentum, but it is difficult to predict the rate of evolution, particularly for low mass (compared to the star) disks which nonetheless still contain enough material to account for the observed planetary system. There is no compelling evidence, from available theories of disk structure and evolution, that the solar nebula must have evolved rapidly and could not have persisted for more than 1 Ma. In considering chronoloically relevant isotopic data for meteorites, we focus on three methodologies: absolute ages by U-Pb/Pb-Pb, and relative ages by short-lived radionuclides (especially Al-26) and by evolution of Sr-87/Sr-86. Two kinds of meteoritic materials-refractory inclusions such as CAIs and differential meteorites (eucrites and augrites) -- appear to have experienced potentially dateable nebular events. In both cases, the most straightforward interpretations of the available data indicate nebular events spanning several Ma. We also consider alternative interpretations, particularly the hypothesis of radically heterogeneous distribution of Al-26, which would avoid these chronological interpretations. The principal impetus for such alternative interpretations seems to be precisely the obviation of the chronological interpretation (i.e., the presumption rather than the inference of a short (less than or equal to 1 Ma) lifetime of the nebula). Astronomical observations of T Tauri stars indicate that the presence of dusty disks is a common if not universal feature, that the disks are massive enough to accomodate a planetary system such as ours, and that at least some persist for 110(exp 7) yr or more. The results are consistent with the time scales inferred from the meteoritic isotopic data. They cannot be considered conclusive with regard to solar nebula time scales, however, in part because it is difficult to relate disk observations to processes that affect meteorites, and in part because the ages assigned for these stars could be wrong by a factor of several in either direction. We conclude that the balance of available evidence favors the view that the nebula existed and was active for at least several Ma. However, because the evidence is not definitive, it is important that the issue be perceived to be an open question, whose answer should be sought rather than presumed.

Podosek, Frank A.↗

The stability of hibonite, melilite and other aluminous phases in silicate melts: Implications for the origin of hibonite-bearing inclusions from carbonaceous chondrites

Phase fields in which hibonite and silicate melt coexist with spinel CaAl4O7, gehlenitic melilite, anorthite or corundum at 1 bar in the system CaO-MgO-Al2O3-SiO2-TiO2 were determined. The hibonites contain up to 1.7 wt% SiO2. For TiO2, the experimentally determined partition coefficients between hibonite and coexisting melt D(sub i)(sup Hib/L), vary from 0.8 to 2.1 and generally decrease with increasing TiO2 in the liquid. Based on Ti partitioning between hibonite and melt, bulk inclusion compositions and hibonite-saturated liquidus phase diagrams, the hibonite in hibonite-poor fluffy Type A inclusions from Allende and at least some hibonite from hibonite-rich inclusions is relict, although much of the hibonite from hibonite-glass spherules probably crystallized metasably from a melt. Bulk compositions for all of these CAIs are consistent with an origin as melite + hibonite + spinel + perovskite phase assembalges that were partially altered and in some cases partially or completely melted. The duration of the melting event was sufficient to remove any Na introduced by the alteration process but frequently insufficient to dissolve all of the original hibonite. Simple thermochemical models developed for meteoritic melilite and hibonite solid solutions were used to obtain equilibration temperatures of hibonite-bearing phase assemblages with vapor. Referenced to 10(exp -3) atm, hibonite + corundum + vapor equilibrated at approximately 1260 C and hibonite + spinel +/- melilite + vapor at 1215 +/- 10 C. If these temperatures reflect condensation in a cooling gas of solar composition, then hibonite +/- corundum condensed first, followed by spinel and then melilite. The position of perovskite within this sequence is uncertain, but it probably began to condense before spinel. This sequence of phase appearances and relative temperatures is generally consistent with observed textures but differs from expectations based on classical condensation calculations in that equilibration temperatures are generally lower than predicted and melilite initially condenses with or even after spinel. Simple thermochemical modes for the substitution of trace elements into the Ca site of meteoritic hibonites suggest that virtually all Eu is divalent in early condensate hibonites but that Eu(2+)/Eu(#+) decreases by a factor of 20 or more during the course of condensation primarily because the ratio is proportional to the partial pressure of Al, which decreases dramatically as aluminous phase condense. The relative sizes of Eu and Yb anomalies in meteoritic hibonites and inclusions may be partly due to this effect.

Beckett, J. R.↗

Out-of-plane properties

This paper summarizes three areas of research which were performed to characterize out-of-plane properties of composite materials. In the first investigation, a series of tests was run to characterize the through-the-thickness tensile strength for a variety of composites that included 2D braids, 2D and 3D weaves, and prepreg tapes. A new test method based on a curved beam was evaluated. Failures were significantly different between the 2D materials and the 3D weaves. The 2D materials delaminated between layers due to out-of-plane tensile stresses while the 3D weaves failed due to the formation of radial cracks between the surface plies caused by high circumferential stresses along the inner radius. The strength of the 2D textile composites did not increase relative to the tapes. Final failure in the 3D weaves was caused by a circumferential crack similar to the 2D materials and occurred at a lower bending moment than in other materials. The early failures in the 3D weaves were caused by radial crack formation rather than a low through-the-thickness strength. The second investigation focused on the development of a standard impact test method to measure impact damage resistance. The only impact tests that currently exist are compression after impact (CAI) tests which incorporate elements of both damage resistance and damage tolerance. A new impact test method is under development which uses a quasi-static indentation (QSI) test to directly measure damage resistance. Damage resistance is quantified in terms of the contact force to produce a unit of damage where a metric for damage may be area in C-scan, depth of residual dent , penetration, damage growth, etc. A final draft of an impact standard that uses a QSI test method will be presented to the ASTM Impact Task Group on impact. In the third investigation, the impact damage resistance behavior of a variety of textile materials was studied using the QSI test method. In this study, the force where large damage initiates was measured and the delamination size as a function of force was determined. The force to initiate large damage was significantly lower in braids and weaves. The delamination diameter - impact forace relationship was quanitfied using a damage resistance parameter, Q(*), which related delamination diameter to imapct force over a range of delamination sizes. Using this Q(*) parameter to rate the materials, the stitched uniweaves, toughened epoxy tapes, and through-the-thickness orthogonal interlock weave were the most damage resistant.

Jackson, Wade C.↗

Evaluation of the impact response of textile composites

An evaluation of the impact damage resistance and impact damage tolerance of stitched and unstitched uniweaves, 2-D braids, and 3-D weaves was conducted. Uniweave laminates were tested at four thicknesses to determine the sensitivity of the tests to this parameter. Several braid and weave parameters were also varied to establish their velocity (large mass) impacts and then loaded in tension or compression to measure residual strength. Experimental results indicate that stitching significantly improves the uniweaves' damage resistance. The 2-D braids and 3-D weaves offered less damage resistance than the stitched materials. Stitching also improved the compression after impact (CAI) and tension after impact (TAI) strengths of the uniweave materials.

Portanova, M. A.↗

Implementation of Hybrid V-Cycle Multilevel Methods for Mixed Finite Element Systems with Penalty

The goal of this paper is the implementation of hybrid V-cycle hierarchical multilevel methods for the indefinite discrete systems which arise when a mixed finite element approximation is used to solve elliptic boundary value problems. By introducing a penalty parameter, the perturbed indefinite system can be reduced to a symmetric positive definite system containing the small penalty parameter for the velocity unknown alone. We stabilize the hierarchical spatial decomposition approach proposed by Cai, Goldstein, and Pasciak for the reduced system. We demonstrate that the relative condition number of the preconditioner is bounded uniformly with respect to the penalty parameter, the number of levels and possible jumps of the coefficients as long as they occur only across the edges of the coarsest elements.

Lai, Chen-Yao G.↗