A Petrov Galerkin finite-element method for interface problems arising in sensitivity computations. (English) Zbl 1087.65078
A discontinuous Petrov Galerkin finite element method is proposed and analyzed to deal with the computation of sensitivity analysis of 1D interface problems. First, to motivate the difficulties arising in the numerical solution of these problems the authors start by introducing a simple example of steady state heat conduction in a thin rod with \( \Omega=[0,1]\) consisting of two materials with the interface at the spatial location \( x=q,\) \(q \in (0,1)\).
It is shown that a weak formulation is necessary to capture the discontinuities of the solution and by means of a Petrov Galerkin method with suitable trial functions they are able to establish the existence of a unique solution and to prove the convergence in suitable spaces.
Further, the paper presents the results of two numerical examples to show the advantage of the proposed technique over a standard finite element approach because it avoids Gibbs phenomena in the discontinuity of the sensitivity across the interface.
It is shown that a weak formulation is necessary to capture the discontinuities of the solution and by means of a Petrov Galerkin method with suitable trial functions they are able to establish the existence of a unique solution and to prove the convergence in suitable spaces.
Further, the paper presents the results of two numerical examples to show the advantage of the proposed technique over a standard finite element approach because it avoids Gibbs phenomena in the discontinuity of the sensitivity across the interface.
Reviewer: Manuel Calvo (Zaragoza)
MSC:
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 34B05 | Linear boundary value problems for ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
Keywords:
1D Elliptic Interface Problems; Petrov Galerkin finite elements; sensitivity analysis; heat conduction; convergence; numerical examplesReferences:
| [1] | Borggaard, J. T.; Burns, J. A., A PDE sensitivity equation method for optimal oerodynamic design, Journal of Computational Physics, 136, 366-384 (1997) · Zbl 0903.76064 |
| [2] | Stanley, L. G., Sensitivity Analysis and Computations for Elliptic Boundary Value Problems, (Ph.D. Thesis (1999), Department of Mathematics, Virginia Tech) · Zbl 0996.65121 |
| [3] | Stewart, D. L., Numerical Methods for Accurate Computation of Design Sensitivities, (Ph.D. Thesis (1998), Department of Mathematics, Virginia Tech: Department of Mathematics, Virginia Tech Blacksburg, VA) · Zbl 1106.76353 |
| [4] | Godfrey, A. G., Using sensitivities for flow analysis, (Borggaard, J.; Burns, J.; Cliff, E.; Schreck, S., Computational Methods for Optimal Design and Control (1998)), 181-196, Birkhauser · Zbl 1041.76539 |
| [5] | Borggaard, J. T.; Pelletier, D.; Turgeon, É., Parametric uncertainty analysis for thermal fluid calculations, Journal of Nonlinear Analysis: Series A, Theory and Methods, 47, 4533-4543 (2001) · Zbl 1042.76540 |
| [6] | Lions, J. L., Controle Optimal des Systèmes Gouvernés par des Équations aux Dérivées Partielles (1969), Dunod: Dunod Blacksburg, VA, English translation, Spring-Verlag, New York, (1971) · Zbl 0179.41801 |
| [7] | Aubin, J. P., Approximation of Elliptic Boundary-Value Problems (1972), Wiley-Interscience: Wiley-Interscience Paris · Zbl 0248.65063 |
| [8] | Crowley, A. B., Numerical solution of alloy solidification problems revisited, (Bossavit, A.; etal., Free Boundary Problems: Applications and Theory, Volume 120 (1985)), 112-131, Pitman · Zbl 0593.35100 |
| [9] | Crumpton, P. I.; Shaw, G. J.; Ware, A. F., Discretization and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients, Journal of Computational Physics, 116, 343-358 (1995) · Zbl 0818.65113 |
| [10] | Hétu, J.-F.; Ilinca, F.; Pelletier, D. P., Process modeling and optimization: Issues and challenges, (Borggaard, J.; Burns, J.; Cliff, E.; Schreck, S., Computational Methods for Optimal Design and Control (1998)), 249-263, Birkhauser · Zbl 1041.93504 |
| [11] | Quarteroni, A.; Valli, A., Domain Decomposition Methods for Partial Differential Equations (1999), Oxford Science: Oxford Science New York · Zbl 0931.65118 |
| [12] | Wloka, J., Partial Differential Equations (1992), Cambridge University Press · Zbl 0623.35006 |
| [13] | Burns, J. A.; Stanley, L. G., A note on the use of transformations in sensitivity computations for elliptic systems, Journal of Mathematical and Computer Modeling, 33, 101-114 (2001) · Zbl 0972.65083 |
| [14] | Babuska, I.; Aziz, A. K., Survey lectures on the mathematical foundations of the finite element method, (Aziz, A. K., The Mathematical Foundation of the Finite Element Method with Applications to Partial Differential Equations (1972), Academic Press: Academic Press Great Britain), 3-363 |
| [15] | Nečas, J., Sur une méthode pour resourdre les equations aux dérivées partielles du type elliptique, voisine de la variationnelle, Ann. Scuola Norm. Sup. Pisa, 16, 305-326 (1962) · Zbl 0112.33101 |
| [16] | Schultz, M. H., Spline Analysis (1973), Prentice-Hall, Inc. · Zbl 0333.41009 |
| [17] | Marchuk, G. I.; Shaidurov, V. V., Difference Methods and Their Extrapolations (1983), Springer-Verlag: Springer-Verlag Englewood Cliffs, NJ · Zbl 0511.65076 |
| [18] | Lasiecka, I.; Triggiani, R., Differential and Algebraic Riccati Equations with Applications to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, Lecture Notes in Control and Information Sciences, Volume 164 (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0754.93038 |
| [19] | Strang, G.; Fix, G., An Analysis of the Finite Element Method (1973), Prentice-Hall: Prentice-Hall New York · Zbl 0278.65116 |
| [20] | Ewing, R.; Li, Z.; Lin, T.; Lin, Y., The immersed finite volume methods for the elliptic interface problems, Math. Comput. Simulation, 50, 1-4, 63-76 (1999) · Zbl 1027.65155 |
| [21] | Leveque, R.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM Journal of Numerical Analysis, 31, 4, 1019-1044 (1994) · Zbl 0811.65083 |
| [22] | Li, Z.; Lin, T.; Wu, X., New Cartesian grid methods for interface problems using the finite element formulation, Numerische Mathematik, 99, 1, 61-98 (2003) · Zbl 1055.65130 |
| [23] | Marini, L. D.; Quarteroni, A., A relaxation procedure for domain decomposition methods using finite elements, Numerische Mathematik, 55, 5, 575-598 (1989) · Zbl 0661.65111 |
| [24] | Stanley, L. G., A sensitivity equation method for molding processes, (Proceedings of the 2000 IEEE International Conference on Control Applications and IEEE International Symposium on Computer-Aided Control Systems Design, IEEE International Conference on Control Applications (2000)), 1-20, ISBN 0-7803-6565-8, (TM3-2) |
| [25] | Bernardi, C.; Pironneau, O., Derivative with respect to discontinuities in the porosity, C. R. Acad. Sci. Paris, Ser. I, 335, 661-666 (2002) · Zbl 1013.76085 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
