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The mixed form of the Cahn–Hilliard equations is discretized by the hybridized discontinuous Galerkin method. For any chemical energy density, existence and uniqueness of the numerical solution is obtained. The scheme is proved to be unconditionally stable. Convergence of the method is obtained by deriving a priori error estimates that are valid for the Ginzburg–Landau chemical energy density and for convex domains. The paper also contains discrete functional tools, namely discrete Agmon and Gagliardo–Nirenberg inequalities, which are proved to be valid in the hybridizable discontinuous Galerkin spaces. [ABSTRACT FROM AUTHOR]

A combined hybrid mixed and hybridizable discontinuous Galerkin method is formulated for the flow and transport equations. Convergence of the method is obtained by deriving optimal a priori error bounds in the L 2 norm in space. Since the velocity in the transport equation depends on the flow problem, the stabilization parameter in the HDG method is a function of the discrete velocity. In addition, a key ingredient in the convergence proof is the construction of a projection that is shown to satisfy optimal approximation bounds. Numerical examples confirm the theoretical convergence rates and show the efficiency of high order discontinuous elements. [ABSTRACT FROM AUTHOR]

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]

The purpose of this work is to propose and analyze a hybridized discontinuous Galerkin (HDG) method for the generalized Boussinesq equations with singular heat source. We use polynomials of order k, k − 1 and k to approximate the velocity, the pressure and the temperature. By introducing Lagrange multipliers for the pressure, the approximate velocity field obtained by the HDG method is shown to be exactly divergence-free and H(div)-conforming. Under a smallness assumption on the problem data and solutions, we prove by Brouwer's fixed point theorem that the discrete system has a solution in two dimensions. If the viscosity and thermal conductivity are further assumed to be positive constants, a priori error estimates with convergence rate O(h) and efficient and reliable a posteriori error estimates are derived. Finally numerical examples illustrate the theoretical analysis and show the performance of the obtained a posteriori error estimator. [ABSTRACT FROM AUTHOR]

We propose a finite element discretization for the steady, generalized Navier-Stokes equations for fluids with shear-dependent viscosity, completed with inhomogeneous Dirichlet boundary conditions and an inhomogeneous divergence constraint. We establish (weak) convergence of discrete solutions as well as a priori error estimates for the velocity vector field and the scalar kinematic pressure. Numerical experiments complement the theoretical findings. [ABSTRACT FROM AUTHOR]

In this work is considered an elliptic problem, referred to as the Ventcel problem, involving a second-order term on the domain boundary (the Laplace--Beltrami operator). A variational formulation of the Ventcel problem is studied, leading to a finite element discretization. The focus is on the construction of high-order curved meshes for the discretization of the physical domain and on the definition of the lift operator, which is aimed at transforming a function defined on the mesh domain into a function defined on the physical one. This lift is defined in such a way as to satisfy adapted properties on the boundary relative to the trace operator. The Ventcel problem approximation is investigated both in terms of geometrical error and of finite element approximation error. Error estimates are obtained both in terms of the mesh order r > 1 and to the finite element degree k > 1, whereas such estimates usually have been considered in the isoparametric case so far, involving a single parameter k = r. The numerical experiments we led in both 2 and 3 dimensions allow us to validate the results obtained and proved on the a priori error estimates depending on the 2 parameters k and r. A numerical comparison is made between the errors using the former lift definition and the lift defined in this work establishing an improvement in the convergence rate of the error in the latter case. [ABSTRACT FROM AUTHOR]

This article investigates the implicit Euler–Galerkin finite element method applied to a cancer invasion model with nonlocal diffusion. This model describes the normal cell density, tumor cell density, excess H+ ion concentration, extracellular matrix, and extracellular matrix active metalloproteinases. The primary goal of this work is to utilize numerical solutions to comprehend cancer invasion and transmission within the patient. The Faedo–Galerkin approximation method is initially used to establish the existence of a weak solution, followed by deriving the a priori error estimates of optimal order. Additionally, the implicit Euler–Galerkin finite element method is employed as a numerical tool for solving the given cancer invasion parabolic system. Furthermore, a priori error bounds and convergence estimates for the fully discrete problem are derived. Finally, numerical simulations are conducted to validate the theoretical conclusions and enhance the understanding of how cancer spreads within the patient. [ABSTRACT FROM AUTHOR]

This paper designs, analyzes, and implements a finite element analysis for the simulation of the nonlinear advection-diffusion model. The key feature of the proposed work is to provide novel a priori error estimates in Bôchner norms L 2 (0 , T ; H 0 1 (Ω)) and L ∞ (0 , T ; H 0 1 (Ω)) simultaneously. To achieve the desired error estimates for both semidiscrete and fully discrete schemes, our idea is to introduce the Ritz projection operator that allows us to enforce the known estimates for P 1 and P 2 finite elements in the formulation of the desired results. Further, this paper establishes the stability, existence, and uniqueness in the precise normed spaces. Finally, we prove the fully discrete error estimates using the Backward Euler and Crank-Nicolson's schemes which are corroborated by numerical experiments to validate the developed theory on various domains for both schemes. To the best of author's knowledge, this is the first finite element simulation that provides the estimates in the L 2 Bôchner norm for any such types of time-dependent problems. [ABSTRACT FROM AUTHOR]

This paper focused on approximating a second-order nonlinear hyperbolic optimal control problem. By introducing a new variable, the hyperbolic equation was converted into two parabolic equations. A second-order fully discrete scheme was obtained by combining the Crank-Nicolson formula with the finite element method. The error estimation for this scheme was derived utilizing the second-order sufficient optimality condition and auxiliary problems. To validate the effectiveness of the fully discrete scheme, a numerical example was presented. [ABSTRACT FROM AUTHOR]

In this work, we study a strain gradient problem involving mixtures. The variational formulation is written as a first-order in time coupled system of parabolic variational equations. An existence and uniqueness result is recalled. Then, we introduce a fully discrete approximation by using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved. Finally, some one- and two-dimensional numerical simulations are performed. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to discuss hp spectral element method for optimal control problem governed by a nonlinear elliptic equation with L 2 -norm constraint for control variable. We then set up its weak formulation and hp spectral element approximation scheme. A priori error estimates of hp spectral element approximation based on some suitable projection operators are proved carefully. Using some properties of projection operators, a posteriori error estimates for both the state and the control approximation under some reasonable assumptions are established rigorously. Such estimates are useful tools, which can be used to construct reliable adaptive spectral element methods for optimal control problems. [ABSTRACT FROM AUTHOR]

An iterative proper orthogonal decomposition (POD) method for a parabolic optimal control problem is investigated in this paper. Firstly, we construct the finite element method, where piecewise linear continuous functions are used for space discretization, and the backward Euler method is used for time discretization. Secondly, we apply the POD method as a model order reduction method to reduce the computation. The different POD basis functions for the state and co-state variables are established by an iterative procedure, which takes the finite element solutions at some time instances as snapshots. A priori error estimates are derived for the state, co-state and control variables, respectively. Finally, numerical experiments are provided to support our theoretical 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]

We consider space-time tracking-type distributed optimal control problems for the wave equation in the space-time domain Q:= \Omega \times (0,T) \subset Rn+1, where the control is assumed to be in the energy space [H1,1 0;,0(Q)]\ast, rather than in L2(Q), which is more common. While the latter ensures a unique state in the Sobolev space H1,1 0;0,(Q), this does not define a solution isomorphism. Hence, we use an appropriate state space X such that the wave operator becomes an isomorphism from X onto [H1,1 0;,0(Q)]\ast. Using space-time finite element spaces of piecewise linear continuous basis functions on completely unstructured but shape regular simplicial meshes, we derive a priori estimates for the error \| \widetilde u\varrho h u\| L2(Q) between the computed space-time finite element solution \widetilde u\varrho h and the target function u with respect to the regularization parameter \varrho, and the space-time finite element mesh size h, depending on the regularity of the desired state u. These estimates lead to the optimal choice \varrho = h2 in order to define the regularization parameter \varrho for a given space-time finite element mesh size h or to determine the required mesh size h when \varrho is a given constant representing the costs of the control. The theoretical results will be supported by numerical examples with targets of different regularities, including discontinuous targets. Furthermore, an adaptive space-time finite element scheme is proposed and numerically analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we consider, from a numerical point of view, a two-temperature poro-thermoelastic problem. The model is written as a coupled linear system of hyperbolic and elliptic partial differential equations. An existence result is proved and energy decay properties are recalled. Then we introduce a fully discrete approximation by using the finite element method and the implicit Euler scheme. Some a priori error estimates are obtained, from which the linear convergence of the approximation is deduced under an appropriate additional regularity. Finally, some numerical simulations are performed to demonstrate the accuracy of the approximation, the decay of the discrete energy and the behaviour of the solution depending on a constitutive parameter. [ABSTRACT FROM AUTHOR]

In this paper, we present a two-grid scheme of fully discrete finite element approximation for optimal control problems governed by parabolic integro-differential equations. The state and co-state variables are approximated by a piecewise linear function and the control variable is discretized by a piecewise constant function. First, we derive the optimal a priori error estimates for all variables. Second, we prove the global superconvergence by using the recovery techniques. Third, we construct a two-grid algorithm and discuss its convergence. In the proposed two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid; additionally, the solution of a linear algebraic system on the fine grid and the resulting solution maintain an asymptotically optimal accuracy. Finally, we present a numerical example to verify the theoretical results.

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]

In this article, we use space-time continuous Galerkin (STCG) method to find the numerical solution for two-dimensional (2D) Burgers' equation. The STCG method differs from conventional finite element methods, both the spatial and temporal variables are discretized by finite element method, thus it can easily obtain the high order accuracy in both time and space directions and the corresponding discrete scheme is unconditionally stable. We demonstrate the existence and uniqueness of numerical solution by using Brouwer's fixed point theorem and give the a priori error estimates without needing to satisfy the limitation of the space-time mesh ratio condition. At last, we provide two numerical examples to confirm the efficiency of the method. Moreover, the numerical experiments reveal that the proposed method can handle the bigger Reynolds number than the existing literatures in the case of the absence of stabilization techniques and has the better stability as well as does not need to impose a limitation on the time step size. [ABSTRACT FROM AUTHOR]

In this paper, we consider, from both analytical and numerical viewpoints, a thermoelastic problem. The so-called MGT model, with two different relaxation parameters, is used for both the displacements and the thermal displacement, leading to a linear coupled system made by two third-order in time partial differential equations. Then, using the theory of linear semi-groups the existence and uniqueness to this problem is proved. If we restrict ourselves to the one-dimensional case, the exponential decay of the energy is obtained assuming some conditions on the constitutive parameters. Then, using the classical finite element method and the implicit Euler scheme, we introduce a fully discrete approximation of a variational formulation of the thermomechanical problem. A main a priori error estimates result is shown, from which we conclude the linear convergence under suitable additional regularity conditions. Finally, we present some one-dimensional numerical simulations to demonstrate the convergence of the fully discrete approximation, the behavior of the discrete energy decay and the dependence on a coupling parameter. [ABSTRACT FROM AUTHOR]

We adopt the integral definition of the fractional Laplace operator and analyze solution techniques for fractional, semilinear, and elliptic optimal control problems posed on Lipschitz polytopes. We consider two strategies of discretization: a semidiscrete scheme where the admissible control set is not discretized and a fully discrete scheme where such a set is discretized with piecewise constant functions. As an instrumental step, we derive error estimates for finite element discretizations of fractional semilinear elliptic partial differential equations (PDEs) on quasi-uniform and graded meshes. With these estimates at hand, we derive error bounds for the semidiscrete scheme and improve the ones that are available in the literature for the fully discrete scheme. [ABSTRACT FROM AUTHOR]

We present a new analysis of finite element methods for partial differential equations over curved domains. In many applications, a change of variables translates a physical Poisson problem over a curved physical domain into a parametric Poisson problem over a polytopal parametric domain. Whilst this change of variables greatly simplifies the geometry and numerical implementations, the coordinate transformation typically features only low regularity. In the parametric Poisson problem, this manifests as rough coefficients and data, which diminish the elliptic regularity, and as roughness of the parametric solution. Our main result addresses how to nevertheless recover high-order finite element convergence rates, the key component being a recently developed broken Bramble-Hilbert lemma. This analysis has numerous applications, where the geometric transformation is either computable or merely a theoretical tool. We propose a simplified technique as easier, more broadly applicable, yet just as powerful as previous isoparametric methods. In particular, we reassess the error analysis of isoparametric finite element methods and prove high-order error estimates for isoparametric FEM even when the physical solution is not continuous. Numerical examples confirm our theoretical predictions. [ABSTRACT FROM AUTHOR]

In this paper we analyze a lowest order virtual element method for the classic load reaction–convection–diffusion problem and the convection–diffusion spectral problem, where the assumptions on the polygonal meshes allow to consider small edges for the polygons. Under well defined seminorms depending on a suitable stabilization for this geometrical approach, we derive the well posedness of the numerical scheme and error estimates for the load problem, whereas for the spectral problem we derive convergence and error estimates fo the eigenvalues and eigenfunctions. We report numerical tests to asses the performance of the small edges on our numerical method for both problems under consideration. [ABSTRACT FROM AUTHOR]

This paper presents and analyzes a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media. We use a first-order time extrapolation, which allows us to solve the equations implicitly and sequentially. We show that the discrete problem is well posed, and obtain a priori error estimates. Our numerical results validate the theoretical results, i.e. the algorithm converges with first order. [ABSTRACT FROM AUTHOR]

In this paper, a mixed finite element method combined with Crank-Nicolson scheme approximation of parabolic optimal control problems with control constraint is investigated. For the state and co-state, the order Raviart-Thomas mixed finite element spaces and Crank-Nicolson scheme are used for space and time discretization, respectively. The variational discretization technique is used for the control variable. We derive optimal priori error estimates for the control, state and co-state. Some numerical examples are presented to demonstrate the theoretical results.

In this work, we study a new thermoelastic model with two temperatures from the numerical point of view. The problem is written as a coupled linear system whose unknowns are the displacement field and the conductivity and thermodynamic temperatures. An existence and uniqueness result recently proved is recalled. Then, a fully discrete approximation is introduced by using the finite element method and the classical implicit Euler scheme. A main a priori error estimates result is proved and, under some appropriate regularity conditions on the continuous solution, we obtain the linear convergence. Finally, some numerical simulations are presented to demonstrate the numerical convergence and the discrete energy decay. [ABSTRACT FROM AUTHOR]

In this paper, we present a two-grid scheme of fully discrete finite element approximation for optimal control problems governed by parabolic integro-differential equations. The state and co-state variables are approximated by a piecewise linear function and the control variable is discretized by a piecewise constant function. First, we derive the optimal a priori error estimates for all variables. Second, we prove the global superconvergence by using the recovery techniques. Third, we construct a two-grid algorithm and discuss its convergence. In the proposed two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid; additionally, the solution of a linear algebraic system on the fine grid and the resulting solution maintain an asymptotically optimal accuracy. Finally, we present a numerical example to verify the theoretical results. [ABSTRACT FROM AUTHOR]

The stationary Navier–Stokes equations for a non-Newtonian incompressible fluid are coupled with the stationary heat equation and subject to Dirichlet-type boundary conditions. The viscosity is supposed to depend on the temperature and the stress depends on the strain through a suitable power law depending on p ∈ (1 , 2) (shear thinning case). For this problem we establish the existence of a weak solution as well as we prove some regularity results both for the Navier–Stokes and the Stokes cases. Then, the latter case with the Carreau power law is approximated through a FEM scheme and some error estimates are obtained. Such estimates are then validated through some two-dimensional numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, a thermomechanical problem involving a viscoelastic Timoshenko beam is analyzed from a numerical point of view. The so-called thermodiffusion effects are also included in the model. The problem is written as a linear system composed of two second-order-in-time partial differential equations for the transverse displacement and the rotational movement, and two first-order-in-time partial differential equations for the temperature and the chemical potential. The corresponding variational formulation leads to a coupled system of first-order linear variational equations written in terms of the transverse velocity, the rotation speed, the temperature and the chemical potential. The existence and uniqueness of solutions, as well as the energy decay property, are stated. Then, we focus on the numerical approximation of this weak problem by using the implicit Euler scheme to discretize the time derivatives and the classical finite element method to approximate the spatial variable. A discrete stability property and some a priori error estimates are shown, from which we can conclude the linear convergence of the approximations under suitable additional regularity conditions. Finally, some numerical simulations are performed to demonstrate the accuracy of the scheme, the behavior of the discrete energy decay and the dependence of the solution with respect to some parameters. [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]

The aim of this work is the numerical homogenization of a parabolic problem with several time and spatial scales using the heterogeneous multiscale method. We replace the actual cell problem with an alternate one, using Dirichlet boundary and initial values instead of periodic boundary and time conditions. Further, we give a detailed a priori error analysis of the fully discretized, i.e., in space and time for both the macroscopic and the cell problem, method. Numerical experiments illustrate the theoretical convergence rates. [ABSTRACT FROM AUTHOR]

This paper proposes a two-grid mixed finite volume element method (TGMFVE) that uses a θ time discrete scheme to solve the Cahn–Hilliard equation. This method is separated into two steps. In the first step, the solution of the Cahn–Hilliard equation can be obtained by using a mixed θ scheme of the finite volume element method on a coarse grid using an iterative algorithm. The second step involves using the linearized mixed θ scheme finite volume element method to solve the equation on a fine grid. The stability analysis of the θ scheme of the two-grid mixed finite volume element method has been performed. The priori error estimation for L 2 norm and H 1 norm is also analyzed. The results of theoretical analysis are confirmed by numerical experiments. The results show that the theoretical results match the actual numerical results. [ABSTRACT FROM AUTHOR]

In this paper, we give pointwise estimates of a Voronoï-based finite volume approximation of the Laplace-Beltrami operator on Voronoï-Delaunay decompositions of the sphere. These estimates are the basis for local error analysis, in the maximum norm, of the approximate solution of the Poisson equation and its gradient. Here, we consider the Voronoï-based finite volume method as a perturbation of the finite element method. Finally, using regularized Green’s functions, we derive quasi-optimal convergence order in the maximum-norm with minimal regularity requirements. Numerical examples show that the convergence is at least as good as predicted. [ABSTRACT FROM AUTHOR]

In this paper, we analyze, from the numerical point of view, two thermo-elastic problems involving the Green–Lindsay theory. The coupling term is different for each case, involving second order or first order spatial derivatives, respectively. The variational formulation leads to a linear coupled system which is written in terms of the velocity and temperature speed. An existence and uniqueness results and the exponential energy decay for the problem with the stronger coupling are recalled. The polynomial energy decay for the weaker coupling is then proved but using the theory of linear semigroups. Then, a fully discrete approximation is introduced using the finite element method and an implicit scheme. A discrete stability property and a main a priori error estimates result are shown, from which we can derive the linear convergence of the approximations. Finally, some numerical simulations are presented to demonstrate the accuracy of the algorithm, the discrete energy decay and the dependence on the relaxation parameter. [ABSTRACT FROM AUTHOR]

The locally modified finite element method, which is introduced in Frei and Richter [SIAM J. Numer. Anal.52 (2014) 2315–2334], is a simple fitted finite element method that is able to resolve weak discontinuities in interface problems. The method is based on a fixed structured coarse mesh, which is then refined into sub-elements to resolve an interior interface. In this work, we extend the locally modified finite element method in two space dimensions to second order using an isoparametric approach in the interface elements. Thereby we need to take care that the resulting curved edges do not lead to degenerate sub-elements. We prove optimal a priori error estimates in the L2-norm and in a discrete energy norm. Finally, we present numerical examples to substantiate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, a mixed finite element method combined with Crank-Nicolson scheme approximation of parabolic optimal control problems with control constraint is investigated. For the state and co-state, the order m = 1 Raviart-Thomas mixed finite element spaces and Crank-Nicolson scheme are used for space and time discretization, respectively. The variational discretization technique is used for the control variable. We derive optimal priori error estimates for the control, state and co-state. Some numerical examples are presented to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

We discretise a tangential tensor field equation using a surface-finite element approach with a penalisation term to ensure almost tangentiality. It is natural to measure the quality of such a discretisation intrinsically, i.e. to examine the tangential convergence behaviour in contrast to the normal behaviour. We show optimal-order convergence with respect to the tangential quantities in particular for an isogeometric penalisation term that is based only on the geometric information of the discrete surface. [ABSTRACT FROM AUTHOR]

The main aim of this article is to analyze mixed finite element method for the second order Dirichlet boundary control problem. We develop both a priori and a posteriori error analysis using the energy space based approach. We obtain optimal order a priori error estimates in the energy norm and L 2 -norm with the help of auxiliary problems. The reliability and the efficiency of proposed a posteriori error estimator is discussed using the Helmholtz decomposition. Numerical experiments are presented to confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we study the parabolic Dirichlet boundary optimal control on complex connected domains. It is well known that both the complex connected domain and the Dirichlet boundary control bring great difficulties to theoretical analysis and numerical calculation. The complex connected domain is a typical non-convex domain, and it is difficult for the traditional numerical method to obtain the same convergence order as the state and adjoint state for the Dirichlet boundary control. The optimal system of the proposed control problem is first obtained by using Lagrange multiplier method. Then, based on the variational form, the Fourier finite volume element method is used to obtain the fully discrete scheme of the optimality system, that is, using Fourier expansion method in azimuthal direction and applying finite volume element method in radial direction respectively, and the Crank-Nicolson scheme in the time direction. Next, we strictly prove the error estimates of state, adjoint state and Dirichlet boundary control by using the variational discretization concept. Finally, the theoretical analysis results and the feasibility and effectiveness of the proposed method are verified by numerical experiments. • It is significant to develop the Fourier finite volume element method for optimal Dirichlet boundary control problems. • We develop and analyze the Fourier finite volume element method for an optimal Dirichlet boundary control problem. • Rigorous error estimates are provided for Dirichlet boundary control, state and adjoint state. • Numerical examples confirm the correctness of the theoretical analysis and the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This article discusses the well-posedness and error analysis of the coupling of finite and boundary elements for interface problems in nonlinear elasticity. It concerns 푝-Laplacian-type Hencky materials with an unbounded stress-strain relation, as they arise in the modelling of ice sheets, non-Newtonian fluids or porous media. We propose a functional analytic framework for the numerical analysis and obtain a priori and a posteriori error estimates for Galerkin approximations to the resulting boundary/domain variational inequality. [ABSTRACT FROM AUTHOR]

In this paper, we analyze, from the numerical point of view, a swelling porous thermo-elastic problem. The so-called second-sound effect is introduced and modeled by using the simplest Maxwell–Cattaneo law. This problem leads to a coupled system which is written by using the displacements of the fluid and the solid, the temperature and the heat flux. The numerical analysis of this problem is performed applying the classical finite element method with linear elements for the spatial approximation and the backward Euler scheme for the discretization of the time derivatives. Then, we prove the stability of the discrete solutions and we provide an a priori error analysis. Finally, some numerical simulations are performed to demonstrate the accuracy of the approximations, the exponential decay of the discrete energy and the dependence on a coupling parameter. [ABSTRACT FROM AUTHOR]

We analyse an energy minimisation problem recently proposed for modelling smectic-A liquid crystals. The optimality conditions give a coupled nonlinear system of partial differential equations, with a second-order equation for the tensor-valued nematic order parameter Q and a fourth-order equation for the scalar-valued smectic density variation u. Our two main results are a proof of the existence of solutions to the minimisation problem, and the derivation of a priori error estimates for its discretisation of the decoupled case (i.e., q = 0) using the C0 C 0 $ {\mathcal{C}}^0$ interior penalty method. More specifically, optimal rates in the H1 and L2 norms are obtained for Q, while optimal rates in a mesh-dependent norm and L2 norm are obtained for u. Numerical experiments confirm the rates of convergence. [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]

An extra-stabilized Morley finite element method (FEM) directly computes guaranteed lower eigenvalue bounds with optimal a priori convergence rates for the bi-Laplacian Dirichlet eigenvalues. The smallness assumption min{λh,λ}h4max ≤184.9570 in 2D (resp., ≤21.2912 in 3D) on the maximal mesh-size hmax makes the computed k th discrete eigenvalue λh≤λ a lower eigenvalue bound for the k th Dirichlet eigenvalue λ. This holds for multiple and clusters of eigenvalues and serves for the localization of the bi-Laplacian Dirichlet eigenvalues, in particular for coarse meshes. The analysis requires interpolation error estimates for the Morley FEM with explicit constants in any space dimension n≥2, which are of independent interest. The convergence analysis in 3D follows the Babuška-Osborn theory and relies on a companion operator for the Morley finite element method. This is based on the Worsey-Farin 3D version of the Hsieh-Clough-Tocher macro element with a careful selection of center points in a further decomposition of each tetrahedron into 12 subtetrahedra. Numerical experiments in 2D support the optimal convergence rates of the extra-stabilized Morley FEM and suggest an adaptive algorithm with optimal empirical convergence rates. [ABSTRACT FROM AUTHOR]

We study the full discretization of a general class of first- and second-order quasilinear wave-type problems with the implicit midpoint rule and a linearized variant thereof. Based on a proof by induction, we prove wellposedness and a rigorous error estimate for both schemes, combining energy techniques, inverse estimates and a linearized fixed-point iteration for the analysis of the nonlinear scheme. To confirm the relevance of the general framework, we derive novel error estimates for the full discretization of two prominent examples from nonlinear physics: the Westervelt equation and the Maxwell equations with Kerr nonlinearity. [ABSTRACT FROM AUTHOR]