An hp‐adaptive Newton‐Galerkin finite element procedure for semilinear boundary value problems

In this paper, we develop an hp‐adaptive procedure for the numerical solution of general, semilinear elliptic boundary value problems in 1d, with possible singular perturbations. Our approach combines both a prediction‐type adaptive Newton method and an hp‐version adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully hp‐adaptive Newton–Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Two‐grid methods for semilinear interface problems

In this article, we consider two‐grid finite element methods for solving semilinear interface problems in d space dimensions, for d = 2 or d = 3. We consider semilinear problems with discontinuous diffusion coefficients, which includes problems containing subcritical, critical, and supercritical nonlinearities. We establish basic quasioptimal a priori error estimates for Galerkin approximations. We then design a two‐grid algorithm consisting of a coarse grid solver for the original nonlinear problem, and a fine grid solver for a linearized problem. We analyze the quality of approximations generated by the algorithm and show that the coarse grid may be taken to have much larger elements than the fine grid, and yet one can still obtain approximation quality that is asymptotically as good as solving the original nonlinear problem on the fine mesh. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

A quadrature finite element method for semilinear second‐order hyperbolic problems

In this work we propose and analyze a fully discrete modified Crank–Nicolson finite element (CNFE) method with quadrature for solving semilinear second‐order hyperbolic initial‐boundary value problems. We prove optimal‐order convergence in both time and space for the quadrature‐modified CNFE scheme that does not require nonlinear algebraic solvers. Finally, we demonstrate numerically the order of convergence of our scheme for some test problems. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Lorenz‐gauged vector potential formulations for the time‐harmonic eddy‐current problem with L∞‐regularity of material properties

In this paper, we consider some Lorenz‐gauged vector potential formulations of the eddy‐current problem for the time‐harmonic Maxwell equations with material properties having only L^∞‐regularity. We prove that there exists a unique solution of these problems, and we show the convergence of a suitable finite element approximation scheme. Moreover, we show that some previously proposed Lorenz‐gauged formulations are indeed formulations in terms of the modified magnetic vector potential, for which the electric scalar potential is vanishing. Copyright © 2007 John Wiley & Sons, Ltd.     

On the subdomain‐Galerkin/least squares method for 2‐ and 3‐D mixed elliptic problems with reaction terms

In this article we apply the subdomain‐Galerkin/least squares method, which is first proposed by Chang and Gunzburger for first‐order elliptic systems without reaction terms in the plane, to solve second‐order non‐selfadjoint elliptic problems in two‐ and three‐dimensional bounded domains with triangular or tetrahedral regular triangulations. This method can be viewed as a combination of a direct cell vertex finite volume discretization step and an algebraic least‐squares minimization step in which the pressure is approximated by piecewise linear elements and the flux by the lowest order Raviart‐Thomas space. This combined approach has the advantages of both finite volume and least‐squares methods. Among other things, the combined method is not subject to the Ladyzhenskaya‐Babus?ka‐Brezzi condition, and the resulting linear system is symmetric and positive definite. An optimal error estimate in the H¹(Ω) × H(div; Ω) norm is derived. An equivalent residual‐type a posteriori error estimator is also given. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 738–751, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10030.     

Robust residual‐ and recovery‐based a posteriori error estimators for a multipoint flux mixed finite element discretization of interface problems

We consider a posteriori error estimation for a multipoint flux mixed finite element method for two‐dimensional elliptic interface problems. Within the class of modified quasi‐monotonically distributed coefficients, we derive a residual‐type a posteriori error estimator of the weighted sum of the scalar and flux errors which is robust with respect to the jumps of the coefficients. Moreover, we develop robust implicit and explicit recovery‐type estimators through gradient recovery in an H(curl)‐conforming finite element space. In particular, we apply a modified L² projection in the implicit recovery procedure so as to reduce the computational cost of the recovered gradient. Numerical experiments confirm the theoretical results.     

A posteriori error estimates of edge residual type of finite element method for nonmonotone quasi‐linear elliptic problems

In this article, we study the edge residual‐based a posteriori error estimates of conforming linear finite element method for nonmonotone quasi‐linear elliptic problems. It is proven that edge residuals dominate a posteriori error estimates. Up to higher order perturbations, edge residuals can act as a posteriori error estimators. The global reliability and local efficiency bounds are established both in H ¹‐norm and L ²‐norm. Numerical experiments are provided to illustrate the performance of the proposed error estimators. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 813–837, 2014     

A mixed‐type Galerkin variational formulation and fast algorithms for variable‐coefficient fractional diffusion equations

We consider the variable‐coefficient fractional diffusion equations with two‐sided fractional derivative. By introducing an intermediate variable, we propose a mixed‐type Galerkin variational formulation and prove the existence and uniqueness of the variational solution over . On the basis of the formulation, we develop a mixed‐type finite element procedure on commonly used finite element spaces and derive the solvability of the finite element solution and the error bounds for the unknown and the intermediate variable. For the Toeplitz‐like linear system generated by discretization, we design a fast conjugate gradient normal residual method to reduce the storage from O(N²) to O(N) and the computing cost from O(N³) to O(NlogN). Numerical experiments are included to verify our theoretical findings. Copyright © 2017 John Wiley & Sons, Ltd.     

Convergence of goal‐oriented adaptive finite element methods for nonsymmetric problems

In this article, we develop convergence theory for a class of goal‐oriented adaptive finite element algorithms for second‐order nonsymmetric linear elliptic equations. In particular, we establish contraction results for a method of this type for Dirichlet problems involving the elliptic operator with A Lipschitz, symmetric positive definite, with b divergence‐free, and with . We first describe the problem class and review some standard facts concerning conforming finite element discretization and error‐estimate‐driven adaptive finite element methods (AFEM). We then describe a goal‐oriented variation of standard AFEM. Following the recent work of Mommer and Stevenson for symmetric problems, we establish contraction and convergence of the goal‐oriented method in the sense of the goal function. Our analysis approach is signficantly different from that of Mommer and Stevenson, combining the recent contraction frameworks developed by Cascon, Kreuzer, Nochetto, and Siebert; by Nochetto, Siebert, and Veeser; and by Holst, Tsogtgerel, and Zhu. We include numerical results, demonstrating performance of our method with standard goal‐oriented strategies on a convection problem. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 479–509, 2016     

Galerkin's finite element formulation of the system of fourth‐order boundary‐value problems

In this article, a Galerkin's finite element approach based on weighted‐residual is presented to find approximate solutions of a system of fourth‐order boundary‐value problems associated with obstacle, unilateral and contact problems. The approach utilizes a piece‐wise cubic approximations utilizing cubic Hermite interpolation polynomials. Numerical studies have shown the superior accuracy and lesser computational cost of the scheme in comparison to cubic spline, non‐polynomial spline and cubic non‐polynomial spline methods. Numerical examples are presented to illustrate the applicability of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1551–1560, 2011     

An implicit–explicit residual error estimator for the coupling of dual‐mixed finite elements and boundary elements in elastostatics

We consider the coupling of dual‐mixed finite elements and boundary elements to solve a mixed Dirichlet–Neumann problem of plane elasticity. We derive an a‐posteriori error estimate that is based on the solution of local Dirichlet problems and on a residual term defined on the coupling interface. The general error estimate does not make use of any special finite element or boundary element spaces. Here the residual term is given in a negative order Sobolev norm. In practical applications, where a certain boundary element subspace is used, this norm can be estimated by weighted local L²‐norms. Copyright © 2001 John Wiley & Sons, Ltd.     

A two‐grid characteristic finite volume element method for semilinear advection‐dominated diffusion equations

A two‐grid finite volume element method, combined with the modified method of characteristics, is presented and analyzed for semilinear time‐dependent advection‐dominated diffusion equations in two space dimensions. The solution of a nonlinear system on the fine‐grid space (with grid size h) is reduced to the solution of two small (one linear and one nonlinear) systems on the coarse‐grid space (with grid size H) and a linear system on the fine‐grid space. An optimal error estimate in H¹ ‐norm is obtained for the two‐grid method. It shows that the two‐grid method achieves asymptotically optimal approximation, as long as the mesh sizes satisfy h = O(H²). Numerical example is presented to validate the usefulness and efficiency of the method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

S‐asymptotically ω‐periodic solutions for semilinear Volterra equations

We study S‐asymptotically ω‐periodic mild solutions of the semilinear Volterra equation u′(t)=(a* Au)(t)+f(t, u(t)), considered in a Banach space X, where A is the generator of an (exponentially) stable resolvent family. In particular, we extend the recent results for semilinear fractional integro‐differential equations considered in (Appl. Math. Lett. 2009; 22:865–870) and for semilinear Cauchy problems of first order given in (J. Math. Anal. Appl. 2008; 343(2): 1119–1130). Applications to integral equations arising in viscoelasticity theory are shown. Copyright © 2010 John Wiley & Sons, Ltd.     

A posteriori error analysis of a cell-centered finite volume method for semilinear elliptic problems

In this paper, we conduct a goal-oriented a posteriori analysis for the error in a quantity of interest computed from a cell-centered finite volume scheme for a semilinear elliptic problem. The a posteriori error analysis is based on variational analysis, residual errors and the adjoint problem. To carry out the analysis, we use an equivalence between the cell-centered finite volume scheme and a mixed finite element method with special choice of quadrature.     

A discontinuous interpolated finite volume approximation of semilinear elliptic optimal control problems

In this article, we describe a discontinuous finite volume method with interpolated coefficients for the numerical approximation of the distributed optimal control problem governed by a class of semilinear elliptic equations with control constraints. The proposed distributed control problem involves three unknown variable: control, state and costate. For the approximation of control, we have adopted three different methodologies: variational discretization, piecewise constant and piecewise linear discretization, while the approximation of state and costate variables is based on discontinuous piecewise linear polynomials. As the resulted scheme is non‐symmetric, optimize‐then‐discretize approach is used to approximate the control problem. Optimal a priori error estimates in suitable natural norms for state, costate and control variables are derived. Moreover, numerical experiments are presented to support the derived theoretical results. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2090–2113, 2017     

A posteriori error estimates in quantities of interest for the finite element heterogeneous multiscale method

We present an “a posteriori” error analysis in quantities of interest for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. The multiscale method is based on a macro‐to‐micro formulation, where the macroscopic physical problem is discretized in a macroscopic finite element space, and the missing macroscopic data are recovered on‐the‐fly using the solutions of corresponding microscopic problems. We propose a new framework that allows to follow the concept of the (single‐scale) dual‐weighted residual method at the macroscopic level in order to derive a posteriori error estimates in quantities of interests for multiscale problems. Local error indicators, derived in the macroscopic domain, can be used for adaptive goal‐oriented mesh refinement. These error indicators rely only on available macroscopic and microscopic solutions. We further provide a detailed analysis of the data approximation error, including the quadrature errors. Numerical experiments confirm the efficiency of the adaptive method and the effectivity of our error estimates in the quantities of interest. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Analysis of two‐grid discretization scheme for semilinear hyperbolic equations by mixed finite element methods

In this paper, the full discrete scheme of mixed finite element approximation is introduced for semilinear hyperbolic equations. To solve the nonlinear problem efficiently, two two‐grid algorithms are developed and analyzed. In this approach, the nonlinear system is solved on a coarse mesh with width H, and the linear system is solved on a fine mesh with width h≪H. Error estimates and convergence results of two‐grid method are derived in detail. It is shown that if we choose in the first algorithm and in the second algorithm, the two‐grid algorithms can achieve the same accuracy of the mixed finite element solutions. Finally, the numerical examples also show that the two‐grid method is much more efficient than solving the nonlinear mixed finite element system directly.     

Two‐level Fourier analysis of multigrid for higher‐order finite‐element discretizations of the Laplacian

In this paper, we employ local Fourier analysis (LFA) to analyze the convergence properties of multigrid methods for higher‐order finite‐element approximations to the Laplacian problem. We find that the classical LFA smoothing factor, where the coarse‐grid correction is assumed to be an ideal operator that annihilates the low‐frequency error components and leaves the high‐frequency components unchanged, fails to accurately predict the observed multigrid performance and, consequently, cannot be a reliable analysis tool to give good performance estimates of the two‐grid convergence factor. While two‐grid LFA still offers a reliable prediction, it leads to more complex symbols that are cumbersome to use to optimize parameters of the relaxation scheme, as is often needed for complex problems. For the purposes of this analytical optimization as well as to have simple predictive analysis, we propose a modification that is “between” two‐grid LFA and smoothing analysis, which yields reasonable predictions to help choose correct damping parameters for relaxation. This exploration may help us better understand multigrid performance for higher‐order finite element discretizations, including for Q₂‐Q₁ (Taylor‐Hood) elements for the Stokes equations. Finally, we present two‐grid and multigrid experiments, where the corrected parameter choice is shown to yield significant improvements in the resulting two‐grid and multigrid convergence factors.     

A recovery‐based error estimator for finite volume methods of interface problems: Conforming linear elements

In this article, we present a recovery‐based a posteriori error estimator for finite volume methods which use the conforming linear trial functions to approximate elliptic interface problems. The reliability and efficiency bounds for the error estimator are established by recovering the flux from a weighted L² projection to H(div) conforming finite element spaces. Numerical experiments are given to support the conclusions. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

A stabilized finite element method for convection–diffusion problems

A stabilized finite element method (FEM) is presented for solving the convection–diffusion equation. We enrich the linear finite element space with local functions chosen according to the guidelines of the residual‐free bubble (RFB) FEM. In our approach, the bubble part of the solution (the microscales) is approximated via an adequate choice of discontinuous bubbles allowing static condensation. This leads to a streamline‐diffusion FEM with an explicit formula for the stability parameter τ_K that incorporates the flow direction, has the capability to deal with problems where there is substantial variation of the Péclet number, and gives the same limit as the RFB method. The method produces the same a priori error estimates that are typically obtained with streamline‐upwind Petrov/Galerkin and RFB. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011