#### Table of Content

15 May 2021, Volume 39 Issue 3
SOURCE TERM IDENTIFICATION WITH DISCONTINUOUS DUAL RECIPROCITY APPROXIMATION AND QUASI-NEWTON METHOD FROM BOUNDARY OBSERVATIONS
2021, 39(3):  311-332.  DOI: 10.4208/jcm.1912-m2019-0121
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This paper deals with discontinuous dual reciprocity boundary element method for solving an inverse source problem. The aim of this work is to determine the source term in elliptic equations for nonhomogenous anisotropic media, where some additional boundary measurements are required. An equivalent formulation to the primary inverse problem is established based on the minimization of a functional cost, where a regularization term is employed to eliminate the oscillations of the noisy data. Moreover, an efficient algorithm is presented and tested for some numerical examples.
WELL-CONDITIONED FRAMES FOR HIGH ORDER FINITE ELEMENT METHODS
Kaibo Hu, Ragnar Winther
2021, 39(3):  333-357.  DOI: 10.4208/jcm.2001-m2018-0078
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The purpose of this paper is to discuss representations of high order C0 finite element spaces on simplicial meshes in any dimension. When computing with high order piecewise polynomials the conditioning of the basis is likely to be important. The main result of this paper is a construction of representations by frames such that the associated L2 condition number is bounded independently of the polynomial degree. To our knowledge, such a representation has not been presented earlier. The main tools we will use for the construction is the bubble transform, introduced previously in[1], and properties of Jacobi polynomials on simplexes in higher dimensions. We also include a brief discussion of preconditioned iterative methods for the finite element systems in the setting of representations by frames.
AN ADAPTIVE TRUST-REGION METHOD FOR GENERALIZED EIGENVALUES OF SYMMETRIC TENSORS
Yuting Chen, Mingyuan Cao, Yueting Yang, Qingdao Huang
2021, 39(3):  358-374.  DOI: 10.4208/jcm.2001-m2019-0017
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For symmetric tensors, computing generalized eigenvalues is equivalent to a homogenous polynomial optimization over the unit sphere. In this paper, we present an adaptive trustregion method for generalized eigenvalues of symmetric tensors. One of the features is that the trust-region radius is automatically updated by the adaptive technique to improve the algorithm performance. The other one is that a projection scheme is used to ensure the feasibility of all iteratives. Global convergence and local quadratic convergence of our algorithm are established, respectively. The preliminary numerical results show the efficiency of the proposed algorithm.
TWO NOVEL GRADIENT METHODS WITH OPTIMAL STEP SIZES
Harry Oviedo, Oscar Dalmau, Rafael Herrera
2021, 39(3):  375-391.  DOI: 10.4208/jcm.2001-m2018-0205
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In this work we introduce two new Barzilai and Borwein-like steps sizes for the classical gradient method for strictly convex quadratic optimization problems. The proposed step sizes employ second-order information in order to obtain faster gradient-type methods. Both step sizes are derived from two unconstrained optimization models that involve approximate information of the Hessian of the objective function. A convergence analysis of the proposed algorithm is provided. Some numerical experiments are performed in order to compare the efficiency and effectiveness of the proposed methods with similar methods in the literature. Experimentally, it is observed that our proposals accelerate the gradient method at nearly no extra computational cost, which makes our proposal a good alternative to solve large-scale problems.
A MIXED VIRTUAL ELEMENT METHOD FOR THE BOUSSINESQ PROBLEM ON POLYGONAL MESHES
Gabriel N. Gatica, Mauricio Munar, Filánder A. Sequeira
2021, 39(3):  392-427.  DOI: 10.4208/jcm.2001-m2019-0187
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In this work we introduce and analyze a mixed virtual element method (mixed-VEM) for the two-dimensional stationary Boussinesq problem. The continuous formulation is based on the introduction of a pseudostress tensor depending nonlinearly on the velocity, which allows to obtain an equivalent model in which the main unknowns are given by the aforementioned pseudostress tensor, the velocity and the temperature, whereas the pressure is computed via a postprocessing formula. In addition, an augmented approach together with a fixed point strategy is used to analyze the well-posedness of the resulting continuous formulation. Regarding the discrete problem, we follow the approach employed in a previous work dealing with the Navier-Stokes equations, and couple it with a VEM for the convection-diffusion equation modelling the temperature. More precisely, we use a mixed-VEM for the scheme associated with the fluid equations in such a way that the pseudostress and the velocity are approximated on virtual element subspaces of ${\Bbb H}$(div) and H1, respectively, whereas a VEM is proposed to approximate the temperature on a virtual element subspace of H1. In this way, we make use of the L2-orthogonal projectors onto suitable polynomial spaces, which allows the explicit integration of the terms that appear in the bilinear and trilinear forms involved in the scheme for the fluid equations. On the other hand, in order to manipulate the bilinear form associated to the heat equations, we define a suitable projector onto a space of polynomials to deal with the fact that the diffusion tensor, which represents the thermal conductivity, is variable. Next, the corresponding solvability analysis is performed using again appropriate fixed-point arguments. Further, Strang-type estimates are applied to derive the a priori error estimates for the components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure. The corresponding rates of convergence are also established. Finally, several numerical examples illustrating the performance of the mixed-VEM scheme and confirming these theoretical rates are presented.
CONVERGENCE OF NUMERICAL SCHEMES FOR A CONSERVATION EQUATION WITH CONVECTION AND DEGENERATE DIFFUSION
R. Eymard, C. Guichard, Xavier Lhébrard
2021, 39(3):  428-452.  DOI: 10.4208/jcm.2002-m2018-0287
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The approximation of problems with linear convection and degenerate nonlinear diffusion, which arise in the framework of the transport of energy in porous media with thermodynamic transitions, is done using a θ-scheme based on the centred gradient discretisation method. The convergence of the numerical scheme is proved, although the test functions which can be chosen are restricted by the weak regularity hypotheses on the convection field, owing to the application of a discrete Gronwall lemma and a general result for the time translate in the gradient discretisation setting. Some numerical examples, using both the Control Volume Finite Element method and the Vertex Approximate Gradient scheme, show the role of θ for stabilising the scheme.
MONOLITHIC MULTIGRID FOR REDUCED MAGNETOHYDRODYNAMIC EQUATIONS
Xiaodi Zhang, Weiying Zheng
2021, 39(3):  453-470.  DOI: 10.4208/jcm.2006-m2020-0071
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In this paper, the monolithic multigrid method is investigated for reduced magnetohydrodynamic equations. We propose a diagonal Braess-Sarazin smoother for the finite element discrete system and prove the uniform convergence of the MMG method with respect to mesh sizes. A multigrid-preconditioned FGMRES method is proposed to solve the magnetohydrodynamic equations. It turns out to be robust for relatively large physical parameters. By extensive numerical experiments, we demonstrate the optimality of the monolithic multigrid method with respect to the number of degrees of freedom.
ERROR ESTIMATES FOR SPARSE OPTIMAL CONTROL PROBLEMS BY PIECEWISE LINEAR FINITE ELEMENT APPROXIMATION
Xiaoliang Song, Bo Chen, Bo Yu
2021, 39(3):  471-492.  DOI: 10.4208/jcm.2003-m2017-0213
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Optimization problems with L1-control cost functional subject to an elliptic partial differential equation (PDE) are considered. However, different from the finite dimensional l1-regularization optimization, the resulting discretized L1-norm does not have a decoupled form when the standard piecewise linear finite element is employed to discretize the continuous problem. A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the L1-norm. In this paper, a new discretized scheme for the L1-norm is presented. Compared to the new discretized scheme for L1-norm with the nodal quadrature formula, the advantages of our new discretized scheme can be demonstrated in terms of the order of approximation. Moreover, finite element error estimates results for the primal problem with the new discretized scheme for the L1-norm are provided, which confirms that this approximation scheme will not change the order of error estimates. To solve the new discretized problem, a symmetric Gauss-Seidel based majorized accelerated block coordinate descent(sGS-mABCD) method is introduced to solve it via its dual. The proposed sGS-mABCD algorithm is illustrated at two numerical examples. Numerical results not only confirm the finite element error estimates, but also show that our proposed algorithm is efficient.