Stabilized Space-Time Finite Element Approximations for a Class of Quasilinear Parabolic Problems

In this talk, we present continuous space-time finite element methods for solving quasilinear parabolic problems. We present a discretization error analysis. We present numerical results which confirm the theoretical convergence rate estimates

National Institute for Global Environmental Change. Final Technical Report 1990-2007

Research conducted by the six NIGEC Regional Centers during recent years is reported. An overview of the NIGEC program from its beginnings provides a description and evaluation of the program's vision, strategy and major accomplishments. The program's purpose was to support academic research on environmental change in regions of the country that had historically received relatively little federal funding. The overall vision of NIGEC may be stated as the performance of academic research on the regional interactions between ecosystems and climate. NIGEC's research presents important evidence on the impacts of climate variability and change, and in some cases adaptability, for a broad range of both managed and unmanaged ecosystems, and has thereby documented significant regional issues on the environmental responses to climate change. NIGEC's research has demonstrated large regional differences in the atmospheric carbon exchange budgets of croplands and forests, that there are significant variations of this exchange on diurnal, synoptic, seasonal and interannual time scales due to atmospheric variability (including temperature, precipitation and cloudiness), and that management practices and past history have predominant effects in grasslands and croplands. It is the mid-latitude forests, however, that have received more attention in NIGEC than any other specific ecosystem, and NIGEC's initiation of and participation in the AmeriFlux program, network of carbon flux measurement sites in North American old-growth forests, is generally considered to be its most significant single accomplishment. By including appendices with complete listings of NIGEC publications, principal investigators and participating institutions, this report may also serve as a useful comprehensive documentation of NIGEC

Mesh Grading in Isogeometric Analysis

International audienceThis paper is concerned with the construction of graded meshes for approximating so-called singular solutions of elliptic boundary value problems by means of multipatch discontinuous Galerkin Isogeo-metric Analysis schemes. Such solutions appear, for instance, in domains with re-entrant corners on the boundary of the computational domain, in problems with changing boundary conditions, in interface problems, or in problems with singular source terms. Making use of the analytic behavior of the solution, we construct the graded meshes in the neighborhoods of such singular points following a multipatch approach. We prove that appropriately graded meshes lead to the same convergence rates as in the case of smooth solutions with approximately the same number of degrees of freedom. Representative numerical examples are studied in order to confirm the theoretical convergence rates and to demonstrate the efficiency of the mesh grading technology in Isogeometric Analysis

National Institute for Global Environmental Change. Final Technical Report 1990-2007

Research conducted by the six NIGEC Regional Centers during recent years is reported. An overview of the NIGEC program from its beginnings provides a description and evaluation of the program's vision, strategy and major accomplishments. The program's purpose was to support academic research on environmental change in regions of the country that had historically received relatively little federal funding. The overall vision of NIGEC may be stated as the performance of academic research on the regional interactions between ecosystems and climate. NIGEC's research presents important evidence on the impacts of climate variability and change, and in some cases adaptability, for a broad range of both managed and unmanaged ecosystems, and has thereby documented significant regional issues on the environmental responses to climate change. NIGEC's research has demonstrated large regional differences in the atmospheric carbon exchange budgets of croplands and forests, that there are significant variations of this exchange on diurnal, synoptic, seasonal and interannual time scales due to atmospheric variability (including temperature, precipitation and cloudiness), and that management practices and past history have predominant effects in grasslands and croplands. It is the mid-latitude forests, however, that have received more attention in NIGEC than any other specific ecosystem, and NIGEC's initiation of and participation in the AmeriFlux program, network of carbon flux measurement sites in North American old-growth forests, is generally considered to be its most significant single accomplishment. By including appendices with complete listings of NIGEC publications, principal investigators and participating institutions, this report may also serve as a useful comprehensive documentation of NIGEC

Αριθμητική επίλυση υπερβολικών διαφορικών εξισώσεων με την μέθοδο των ασυνεχών πεπερασμένων στοιχείων

Στην εργασία που ακολουθεί παρουσιάζουμε κυρίως αριθμητικές μεθόδους για την επίλυση γραμμικών και μη-γραμμικών υπερβολικών διαφορικών εξισώσεων, οι οποίες προκύπτουν από την εφαρμογή της αριθμητικής μεθόδου των ασυνεχών πεπερασμένων στοιχείων (ΑΠΣ). Παρουσιάζουμε επίσης αριθμητικές μεθόδους ΑΠΣ για συγγενή προβλήματα, όπως τις εξισώσεις Navier-Stokes, τις εξισώσεις Maxwell και τις εξισώσεις μαγνητοϋδροδυναμικής. Το χωρικό διακριτό ανάλογο των μερικών διαφορικών εξισώσεων (ΜΔΕ) προκύπτει αν διαμερίσουμε το υπολογιστικό χωρίο σε στοιχεία και ορίσουμε κατάλληλους χώρους πεπερασμένης διάστασης στους οποίους ανήκει η προσεγγιστική λύση. Η κύρια διαφορά μεταξύ της μεθόδου των κλασσικών (συνεχών) πεπερασμένων στοιχείων και της μεθόδου των ΑΠΣ, είναι ότι στην μέθοδο των ΑΠΣ δεν απαιτείται συνθήκη συνέχειας για την αριθμητική λύση στις πλευρές των στοιχείων. Η 'επικοινωνία' μεταξύ των γειτονικών στοιχείων επιτυγχάνεται με την επίλυση ενός προβλήματος ασυνέχειας, όπου υπεισέρχεται η αριθμητική ροή (ή Riemann solver) στα σύνορα των στοιχείων. 'Ετσι μπορεί να ειπωθεί ότι η μέθοδος των ΑΠΣ μοιάζει με την μέθοδο των πεπερασμένων όγκων. Η αριθμητική λύση τοπικά σε κάθε στοιχείο είναι ένα πολυώνυμο και η τάξη ακρίβειας της μεθόδου καθορίζεται από τον βαθμό του πολυωνύμου αυτού. Η επίλυση του παραγόμενου συστήματος συνήθων διαφορικών εξισώσεων που προκύπτει από τη διακριτοποίηση του χωρικού προβλήματος ως προς το χρόνο, γίνεται με άμεσες μεθόδους Runge-Kutta τρίτης τάξης. Οι κατηγορίες των μερικών διαφορικών εξισώσεων που επιλύσαμε είναι: 1. Γραμμικές εξισώσεις του Euler (προβλήματα αεροακουστικής), 2. Μη γραμμικές υπερβολικές εξισώσεις Euler (προβλήματα συμπιεστής ροής), 3. Εξισώσεις Navier-Stokes (προβλήματα συμπιεστής ροής με ιξώδες), 4. Εξισώσεις Maxwell (προβλήματα ηλεκτρομαγνητισμού), 5. Εξισώσεις μαγνητοϋδροδυναμικής (MHD) (προβλήματα συμπιεστής ροής υπό την παρουσία μαγνητικού πεδίου). Προσπαθήσαμε να επιλύσουμε τα κλασσικότερα προβλήματα δοκιμής που αναφέρονται στην βιβλιογραφία για κάθε κατηγορία εξισώσεων, ώστε να είναι δυνατή η εξαγωγή τεκμηριωμένων συμπερασμάτων τα οποία αφορούν την αποτελεσματικότητα της μεθόδου των ΑΠΣ για την αριθμητική επίλυση των παραπάνω διαφορικών εξισώσεων

High-order discontinuous Galerkin discretizations for computational aeroacoustics in complex domains

Discontinuous Galerkin discretization is applied for the numerical solution of the linearized Euler equations governing propagation of small-amplitude acoustic disturbances. Second-, fourth-, and sixth-order-accurate numerical solutions are obtained for domain discretizations with triangular elements. In addition, discretizations with quadrilateral unstructured meshes are obtained with second- and fourth-order accuracy. Detailed comparisons with exact solutions and accuracy tests are carried out with results obtained for a pressure pulse propagation and reflection from a solid surface. It is demonstrated that for triangular meshes the numerical solutions show grid convergence according to the order of accuracy used for the discretization. The versatility of the method to obtain accurate predictions in complex geometries is demonstrated with the results obtained for sound scattering from a single cylinder and two cylinders. Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved