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Minimum Residual and Least Squares Finite Element Methods
Computers and Mathematics with Applications 68 (2014) 1479
Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Editorial
Minimum Residual and Least Squares Finite Element Methods
Certain numerical solutions of Partial Differential Equations (PDEs) are couched as a solution of an unconstrained minimization problem for a suitably defined residual of the governing equations. Minimization of this residual over a finitedimensional subspace of the trial space defines the approximate solution. Classical least-squares finite elements are an example of minimum residual methods. There, the PDE is cast into an operator that take values in an L2 space and the residual is measured in the L2 norm as well. For linear problems, the problem reduces to the minimization of a quadratic functional and parallels the classical Ritz approach. The residual can be interpreted as a special energy norm of the discretization error. It provides a natural basis for a posteriori error control and adaptivity. The resulting stiffness matrix is always Hermitian and positive definite. Perhaps the most important property is that the method exhibits no preasymptotic behavior, i.e., one can always start with a very coarse mesh aligned with the problem’s geometry. This is in contrast with classical Bubnov-Galerkin methods, including stabilized methods, that are frequently stable only for ‘‘sufficiently fine’’ meshes. Classical least squares methods have recently been complemented with a range of alternatives in which the residuals are measured in weaker, dual norms. Such methods offer significant additional computational and theoretical advantages over traditional least-squares. Of particular note here are formulations such as discrete ‘‘negative-norm’’ least-squares, the LL∗ methods of Manteuffel, McCormick and Cai, and the discontinuous Petrov Galerkin (DPG) method of Demkowicz and Gopalakrishnan. The DPG method minimizes the residual in a dual norm (like the negative-norm least-squares) but utilizes interface variables and broken test spaces to enable the dual norm computation by an approximate element-wise inversion of Riesz map. The method can be interpreted as a scheme with a space of (approximately) optimal test functions computed on the fly (hence the name). This methodology offers a range of norms in which the residual can be minimized. It also allows for the construction of robust (uniformly stable) discretizations for parameter-dependent and singular-perturbation problems like convection-dominated diffusion, wave propagation, nearly incompressible materials, thin-walled structures etc. The special issue contains contributions from several participants of the ICES/USACM work-shop (with the same name) that took place at Austin in November 4-6, 2013. With 45 attendees and 31 presentations, the workshop aimed at connecting representatives of least squares and DPG communities to facilitate communication and exchange of ideas. For further details, see https://sites.google.com/site/workshoplmr/ for more information and copies of presentations.
Guest Editors Pavel Bochev Leszek Demkowicz Jay Gopalakrishnan Max Gunzburger Available online 19 November 2014
http://dx.doi.org/10.1016/j.camwa.2014.11.005 0898-1221/© 2014 Elsevier Ltd. All rights reserved.
Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Editorial
Minimum Residual and Least Squares Finite Element Methods
Certain numerical solutions of Partial Differential Equations (PDEs) are couched as a solution of an unconstrained minimization problem for a suitably defined residual of the governing equations. Minimization of this residual over a finitedimensional subspace of the trial space defines the approximate solution. Classical least-squares finite elements are an example of minimum residual methods. There, the PDE is cast into an operator that take values in an L2 space and the residual is measured in the L2 norm as well. For linear problems, the problem reduces to the minimization of a quadratic functional and parallels the classical Ritz approach. The residual can be interpreted as a special energy norm of the discretization error. It provides a natural basis for a posteriori error control and adaptivity. The resulting stiffness matrix is always Hermitian and positive definite. Perhaps the most important property is that the method exhibits no preasymptotic behavior, i.e., one can always start with a very coarse mesh aligned with the problem’s geometry. This is in contrast with classical Bubnov-Galerkin methods, including stabilized methods, that are frequently stable only for ‘‘sufficiently fine’’ meshes. Classical least squares methods have recently been complemented with a range of alternatives in which the residuals are measured in weaker, dual norms. Such methods offer significant additional computational and theoretical advantages over traditional least-squares. Of particular note here are formulations such as discrete ‘‘negative-norm’’ least-squares, the LL∗ methods of Manteuffel, McCormick and Cai, and the discontinuous Petrov Galerkin (DPG) method of Demkowicz and Gopalakrishnan. The DPG method minimizes the residual in a dual norm (like the negative-norm least-squares) but utilizes interface variables and broken test spaces to enable the dual norm computation by an approximate element-wise inversion of Riesz map. The method can be interpreted as a scheme with a space of (approximately) optimal test functions computed on the fly (hence the name). This methodology offers a range of norms in which the residual can be minimized. It also allows for the construction of robust (uniformly stable) discretizations for parameter-dependent and singular-perturbation problems like convection-dominated diffusion, wave propagation, nearly incompressible materials, thin-walled structures etc. The special issue contains contributions from several participants of the ICES/USACM work-shop (with the same name) that took place at Austin in November 4-6, 2013. With 45 attendees and 31 presentations, the workshop aimed at connecting representatives of least squares and DPG communities to facilitate communication and exchange of ideas. For further details, see https://sites.google.com/site/workshoplmr/ for more information and copies of presentations.
Guest Editors Pavel Bochev Leszek Demkowicz Jay Gopalakrishnan Max Gunzburger Available online 19 November 2014
http://dx.doi.org/10.1016/j.camwa.2014.11.005 0898-1221/© 2014 Elsevier Ltd. All rights reserved.