IMA Journal of Numerical Analysis
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Published By Oxford University Press

1464-3642, 0272-4979

Author(s):  
Sergio Amat ◽  
David Levin ◽  
Juan Ruiz-Álvarez

Abstract Given values of a piecewise smooth function $f$ on a square grid within a domain $[0,1]^d$, $d=2,3$, we look for a piecewise adaptive approximation to $f$. Standard approximation techniques achieve reduced approximation orders near the boundary of the domain and near curves of jump singularities of the function or its derivatives. The insight used here is that the behavior near the boundaries, or near a singularity curve, is fully characterized and identified by the values of certain differences of the data across the boundary and across the singularity curve. We refer to these values as the signature of $f$. In this paper, we aim at using these values in order to define the approximation. That is, we look for an approximation whose signature is matched to the signature of $f$. Given function data on a grid, assuming the function is piecewise smooth, first, the singularity structure of the function is identified. For example, in the two-dimensional case, we find an approximation to the curves separating between smooth segments of $f$. Secondly, simultaneously, we find the approximations to the different segments of $f$. A system of equations derived from the principle of matching the signature of the approximation and the function with respect to the given grid defines a first stage approximation. A second stage improved approximation is constructed using a global approximation to the error obtained in the first stage approximation.


Author(s):  
Omar Lakkis ◽  
Amireh Mousavi

Abstract We propose a least-squares method involving the recovery of the gradient and possibly the Hessian for elliptic equation in nondivergence form. As our approach is based on the Lax–Milgram theorem with the curl-free constraint built into the target (or cost) functional, the discrete spaces require no inf-sup stabilization. We show that standard conforming finite elements can be used yielding a priori and a posteriori convergence results. We illustrate our findings with numerical experiments with uniform or adaptive mesh refinement.


Author(s):  
Roberto Andreani ◽  
Ariel R Velazco ◽  
Alberto Ramos ◽  
Ademir A Ribeiro ◽  
Leonardo D Secchin

Abstract Augmented Lagrangian (AL) algorithms are very popular and successful methods for solving constrained optimization problems. Recently, global convergence analysis of these methods has been dramatically improved by using the notion of sequential optimality conditions. Such conditions are necessary for optimality, regardless of the fulfillment of any constraint qualifications, and provide theoretical tools to justify stopping criteria of several numerical optimization methods. Here, we introduce a new sequential optimality condition stronger than previously stated in the literature. We show that a well-established safeguarded Powell–Hestenes–Rockafellar (PHR) AL algorithm generates points that satisfy the new condition under a Lojasiewicz-type assumption, improving and unifying all the previous convergence results. Furthermore, we introduce a new primal–dual AL method capable of achieving such points without the Lojasiewicz hypothesis. We then propose a hybrid method in which the new strategy acts to help the safeguarded PHR method when it tends to fail. We show by preliminary numerical tests that all the problems already successfully solved by the safeguarded PHR method remain unchanged, while others where the PHR method failed are now solved with an acceptable additional computational cost.


Author(s):  
K Brenner ◽  
R Masson ◽  
E H Quenjel ◽  
J Droniou

Abstract This work proposes a finite volume scheme for two-phase Darcy flow in heterogeneous porous media with different rock types. The fully implicit discretization is based on cell-centered, as well as face-centered degrees of freedom in order to capture accurately the nonlinear transmission conditions at different rock type interfaces. These conditions play a major role in the flow dynamics. The scheme is formulated with natural physical unknowns, and the notion of global pressure is only introduced to analyze its stability and convergence. It combines a two-point flux approximation of the gradient normal fluxes with a Hybrid Upwinding approximation of the transport terms. The convergence of the scheme to a weak solution is established taking into account the discontinuous capillary pressure at different rock type interfaces and the degeneracy of the phase mobilities. Numerical experiments show the additional robustness of the proposed discretization compared with the classical Phase Potential Upwinding approach.


Author(s):  
Adrian M Ruf

Abstract We prove that adapted entropy solutions of scalar conservation laws with discontinuous flux are stable with respect to changes in the flux under the assumption that the flux is strictly monotone in $u$ and the spatial dependency is piecewise constant with finitely many discontinuities. We use this stability result to prove a convergence rate for the front tracking method—a numerical method that is widely used in the field of conservation laws with discontinuous flux. To the best of our knowledge, both of these results are the first of their kind in the literature on conservation laws with discontinuous flux. We also present numerical experiments verifying the convergence rate results and comparing numerical solutions computed with the front tracking method to finite volume approximations.


Author(s):  
Andreas Dedner ◽  
Alice Hodson

Abstract We present a class of nonconforming virtual element methods for general fourth-order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth-order problems with varying coefficients. We also discuss fourth-order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth-order problems.


Author(s):  
Stefan Güttel ◽  
John W Pearson

Abstract We devise a method for nonlinear time-dependent partial-differential-equation-constrained optimization problems that uses a spectral-in-time representation of the residual, combined with a Newton–Krylov method to drive the residual to zero. We also propose a preconditioner to accelerate this scheme. Numerical results indicate that this method can achieve fast and accurate solution of nonlinear problems for a range of mesh sizes and problem parameters. The numbers of outer Newton and inner Krylov iterations required to reach the attainable accuracy of a spatial discretization are robust with respect to the number of collocation points in time and also do not change substantially when other problem parameters are varied.


Author(s):  
Sören Bartels

Abstract We consider variational problems that model the bending behavior of curves that are constrained to belong to given hypersurfaces. Finite element discretizations of corresponding functionals are justified rigorously via $\varGamma $-convergence. The stability of semi-implicit discretizations of gradient flows is investigated, which provide a practical method to determine stationary configurations. A particular application of the considered models arises in the description of conical sheet deformations.


Author(s):  
Pieter Lietaert ◽  
Karl Meerbergen ◽  
Javier Pérez ◽  
Bart Vandereycken

Abstract We present a method for solving nonlinear eigenvalue problems (NEPs) using rational approximation. The method uses the Antoulas–Anderson algorithm (AAA) of Nakatsukasa, Sète and Trefethen to approximate the NEP via a rational eigenvalue problem. A set-valued variant of the AAA algorithm is also presented for building low-degree rational approximations of NEPs with a large number of nonlinear functions. The rational approximation is embedded in the state-space representation of a rational polynomial by Su and Bai. This procedure perfectly fits the framework of the compact rational Krylov methods (CORK and TS-CORK), allowing solve large-scale NEPs to be efficiently solved. One advantage of our method, compared to related techniques such as NLEIGS and infinite Arnoldi, is that it automatically selects the poles and zeros of the rational approximations. Numerical examples show that the presented framework is competitive with NLEIGS and usually produces smaller linearizations with the same accuracy but with less effort for the user.


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