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We study the linear finite element approximation of piezoelectric unilateral contact problem in a curved and smooth boundary domain. The unilateral contact conditions will be weakly imposed by the penalty method. We derive error estimates which depend on the penalty parameter and the mesh size. In fact, under regularity of the solution, we prove some convergence rate results. [ABSTRACT FROM AUTHOR]

This paper concentrates on a priori error estimates of two monolithic schemes for Biot's consolidation model based on the three‐field formulation introduced by Oyarzúa et al. (SIAM J Numer Anal, 2016). The spatial discretizations are based on the Taylor–Hood finite elements combined with Lagrange elements for the three primary variables. We employ two different schemes to discretize the time domain. One uses the backward Euler method, and the other applies the combination of the backward Euler and Crank‐Nicolson methods. A priori error estimates show that both schemes are unconditionally convergent with optimal error orders. Detailed numerical experiments are presented to validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

The linear elasticity problem has important applications in structural analysis and engineering design. It is shown that the classical Crouzeix–Raviart finite element method is unstable to solve the linear elasticity eigenvalue problem with the pure traction boundary condition. In this paper, first we propose a new scheme with the stabilization terms. Then we prove the a priori error estimates for approximate eigenpairs. Finally, we carry out numerical experiments using the stabilized scheme. Theoretical analysis and numerical results show that this scheme is locking free and efficient. [ABSTRACT FROM AUTHOR]

A semi‐Lagrangian unified finite element method is investigated for solving time‐dependent coupled Darcy‐transport problems. In this method, the modified method of characteristics is combined with a Galerkin finite element discretization allowing the same finite element space to be used for all solutions of the problem including the pressure, velocity and concentration. Convergence and stability of the method are also analyzed in this study and error estimates in the L2$$ {L}^2 $$‐norm are established for the numerical solutions. For the time integration we consider a second‐order implicit method and discrete error estimates are provided. Due to the Lagrangian treatment of convection terms in the considered problems, the conventional Courant‐Friedrichs‐Lewy condition is relaxed and the time truncation errors are reduced in the diffusion‐reaction part. Numerical results are presented for several test examples to demonstrate the reliability and efficiency of the proposed method. The computed results support our expectations for an accurate and highly efficient semi‐Lagrangian unified finite element method for time‐dependent coupled Darcy‐transport problems. [ABSTRACT FROM AUTHOR]

We investigate in the article the L2‐norm state constrained control problem with first bi‐harmonic equation. First, the optimality conditions of the control problem are derived carefully, and spectral discretization of the problem is established. Based on the property of projection operator, we establish a priori error analysis for control variable, state, and adjoint state variable. Furthermore, the efficient projected gradient algorithm is proposed, and the numerical results we obtained verify the analytical results that it can provide high order accuracy and fast convergence rate. [ABSTRACT FROM AUTHOR]

Summary: In this paper, BDF2 schemes combined with the finite element method are proposed to solve control constrained optimal parameter control problems governed by bilinear parabolic equations, where the control variable acts as a reaction coefficient. The discretize‐then‐optimize (DO) approach is adopted. First, the state equation is discretized and two different discrete formulas are utilized for the objective functional. Then, by using the Lagrange multiplier method, two fully discrete first‐order necessary optimality systems are deduced and named as DO1 and DO2 systems, respectively. It can be found that the optimize‐then‐discretize (OD) and discretize‐then‐optimize (DO) approaches coincide in the case of the DO2 system. Furthermore, a priori error estimates are derived for the state, co‐state and control variables of the DO2 system, which show that the DO2 system has second‐order accuracy in time. Finally, three numerical examples are presented to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this work, we study, from the numerical point of view, a type III thermoelastic model with double porosity. The thermomechanical problem is written as a linear system composed of hyperbolic partial differential equations for the displacements and the two porosities, and a parabolic partial differential equation for the thermal displacement. An existence and uniqueness result is recalled. Then, we perform its a priori error numerical analysis approximating the resulting variational problem by using the finite element method and the implicit Euler scheme. The linear convergence of the algorithm is derived under suitable additional regularity conditions. Finally, some numerical simulations are shown to demonstrate the accuracy of the approximations and the dependence of the solution on a coupling coefficient. [ABSTRACT FROM AUTHOR]

Solutions of boundary value problems in three‐dimensional domains with edges may exhibit singularities which are known to influence both the accuracy of the finite element solutions and the rate of convergence in the error estimates. This paper considers boundary value problems for the Poisson equation on typical domains Ω ⊂ ℝ3 with edge singularities and presents, on the one hand, explicit computational formulas for the flux intensity functions. On the other hand, it proposes and analyzes a nonconforming finite element method on regular meshes for the efficient treatment of the singularities. The novelty of the present method is the use of the explicit formulas for the flux intensity functions in defining a postprocessing procedure in the finite element approximation of the solution. A priori error estimates in H1(Ω) show that the present algorithm exhibits the same rate of convergence as it is known for problems with regular solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the local discontinuous Galerkin (LDG) methods for two‐dimensional nonlinear second‐order elliptic problems of the type uxx + uyy = f(x, y, u, ux, uy), in a rectangular region Ω with classical boundary conditions on the boundary of Ω. Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L2 norm achieve optimal (p + 1)th order convergence, when tensor product polynomials of degree at most p are used. Moreover, we prove that the gradient of the LDG solution is superclose with order p + 1 toward the gradient of Gauss–Radau projection of the exact solution. The results are valid in two space dimensions on Cartesian meshes using tensor product polynomials of degree p ≥ 1, and for both mixed Dirichlet–Neumann and periodic boundary conditions. Preliminary numerical experiments indicate that our theoretical findings are optimal. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, the authors present finite element analysis and approximation of Burgers'-Fisher equation. Existence and uniqueness of weak solution is proved by Galerkin's finite element method for non-smooth initial data. Next, a priori error estimates of semi-discrete solution in [ABSTRACT FROM AUTHOR]

In this paper, we study, from a numerical point of view, an osseointegrated model depending on the density of the osteogenic cells and osteoblasts and one generic growth factor. The problem is written as a coupled system of nonlinear partial differential equations and an ordinary differential equation. The variational formulation leads to two coupled parabolic nonlinear variational equations. Then, its numerical approximation is introduced by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. A priori error estimates are then proved, from which the linear convergence is obtained under suitable additional regularity conditions. Finally, some numerical simulations are presented to show the accuracy of the approximation and the behaviour of the solution. [ABSTRACT FROM AUTHOR]

In this article, we develop and analyze a newrecovery-based a posteriori error estimator for the discontinuous Galerkin (DG) method for nonlinear hyperbolic conservation laws on Cartesian grids, when the upwind flux is used. We prove, under some suitable initial and boundary discretizations, that the L2-norm of the solution is of order p +1, when tensor product polynomials of degree at most p are used. We further propose a very simple derivative recovery formula which gives a superconvergent approximation to the directional derivative. The order of convergence is showed to be p + 1/2. We use our derivative recovery result to develop a robust recovery-type a posteriori error estimator for the directional derivative approximation which is based on an enhanced recovery technique. The proposed error estimators of the recovery-type are easy to implement, computationally simple, asymptotically exact, and are useful in adaptive computations. Finally, we show that the proposed recovery-type a posteriori error estimates, at a fixed time, converge to the true errors in the L2-norm under mesh refinement. The order of convergence is proved to be p + 1/2. Our theoretical results are valid for piecewise polynomials of degree p ≥ 1 and under the condition that each component, |f¹i (u)|, i = 1, 2, of the flux function possesses a uniform positive lower bound. Several numerical examples are provided to support our theoretical results and to showthe effectiveness of our recovery-based a posteriori error estimator. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1224-1265, 2017. [ABSTRACT FROM AUTHOR]

We investigate a time-domain Galerkin boundary element method for the wave equation outside a Lipschitz obstacle in an absorbing half-space. A priori estimates are presented for both closed surfaces and screens, and we discuss the relevant properties of anisotropic Sobolev spaces and the boundary integral operators between them. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This article proposes and analyzes a C0-weak Galerkin (WG) finite element method for solving the biharmonic equation in two-dimensional and three-dimensional. The new WG method uses continuous piecewise-polynomial approximations of degree [ABSTRACT FROM AUTHOR]

This paper deals with the analysis of a new augmented mixed finite element method in terms of vorticity, velocity, and pressure, for the Brinkman problem with nonstandard boundary conditions. The approach is based on the introduction of Galerkin least-squares terms arising from the constitutive equation relating the aforementioned unknowns and from the incompressibility condition. We show that the resulting augmented bilinear form is continuous and elliptic, which, thanks to the Lax-Milgram theorem, and besides proving the well-posedness of the continuous formulation, ensures the solvability and stability of the Galerkin scheme with any finite element subspace of the continuous space. In particular, Raviart-Thomas elements of any order [ABSTRACT FROM AUTHOR]

In this article, we investigate the L ∞( L2) -error estimates of the semidiscrete expanded mixed finite element methods for quadratic optimal control problems governed by hyperbolic integrodifferential equations. The state and the costate are discretized by the order k Raviart-Thomas mixed finite element spaces, and the control is approximated by piecewise polynomials of order k( k ≥ 0). We derive error estimates for both the state and the control approximation. Numerical experiments are presented to test the theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 [ABSTRACT FROM AUTHOR]

In this article the ideas in Wang et al. [SIAM J Numec Anal 48 (2010), 708-73] are extended to solve the double obstacle problem using discontinuous Galerkin methods. A priori error estimates are established for these methods, which reach optimal order for linear elements. We present a test example, and the numerical results on the convergence order match the theoretical prediction. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013 [ABSTRACT FROM AUTHOR]

We consider a scalar wave equation with nonseparable spatial scales. If the solution of the wave equation smoothly depends on some global fields, then we can utilize the global fields to construct multiscale finite element basis functions. We present two finite element approaches using the global fields: partition of unity method and mixed multiscale finite element method. We derive a priori error estimates for the two approaches and theoretically investigate the relation between the smoothness of the global fields and convergence rates of the approximations for the wave equation. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011 [ABSTRACT FROM AUTHOR]

The finite-element (FE) approximation of linear elliptic eigenvalue problems is considered. An analysis based on a number of known estimates leads to the simple formula M=r0ℇd/(2p)N relating the total number of degrees of freedom N, the maximum relative error level ℇ desired for the eigenvalues, and the number of ‘reliable’ modes M. (Here d is the spatial dimension and p is the polynomial degree of the FE space.) Moreover, a rough estimate for the numerical value of the constant r0 for a given application is found. This result supports a well-known rule of thumb. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]