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This article introduces a new kind of multigrid approach for semilinear elliptic problems, which is based on the symmetric interior penalty discontinuous Galerkin (SIPDG) method. We first give an optimal error estimate of the SIPDG method for the problem. Then, we design a type of multigrid method, which is called the multilevel correction method, and derive a priori error estimates. The primary idea of this method is to take the solution of the semilinear problem and utilize it to establish a sequence of solutions for associated linear boundary value problem on discontinuous finite element spaces and a newly defined low dimensional augmented subspace. Lastly, numerical experiments are offered to confirm the suggested method's precision and effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, we study the mixed virtual element approximation to an elliptic optimal control problem with boundary observations. The objective functional of this type of optimal control problem contains the outward normal derivatives of the state variable on the boundary, which reduces the regularity of solutions to the optimal control problems. We construct the mixed virtual element discrete scheme and derive an a priori error estimate for the optimal control problem based on the variational discretization for the control variable. Numerical experiments are carried out on different meshes to support our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate virtual element discretization of a semilinear elliptic optimal control problem with pointwise Neumann boundary control constraint. The virtual element discrete scheme is constructed based on virtual element discretization of the state equation and variational discretization of the control variable. By utilizing first-order and second-order optimality conditions along with auxiliary problems, we derive a priori error estimates for the state, adjoint state, and control variable in H 1 and L 2 norms. Theoretical results are confirmed by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we design and analyze the conforming and nonconforming virtual element methods for the Signorini problem. Under some regularity assumptions, we prove optimal order a priori error estimates in the energy norm for both two numerical schemes. Extensive numerical tests are presented, verifying the theory and exploring unknown features. [ABSTRACT FROM AUTHOR]

In this paper, we consider a finite element approximation of an optimal control problem governed by a time fractional wave equation. We first prove the well-posedness and regularity of the optimal control problem. Furthermore, we discuss the fully discrete scheme and analyze its stability and a priori error estimate. Numerical experiments are given to illustrate the theoretical findings. • The weak solution to the time fractional wave equation is studied under low regular data. • The well-posedness and regularity of the optimal control problem are proven. • Numerical schemes exist the first-order and second-order accuracy in time and space respectively. [ABSTRACT FROM AUTHOR]

In this paper we investigate the lowest-order nonconforming virtual element approximation of optimal control problem governed by Stokes equations. Based on the lowest-order virtual element approximation of the state equation and variational discretization of the control variable, we build up the virtual element discrete scheme and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L 2 and H 1 norms are derived. Moreover, the presented method can also handle general polygonal meshes with arbitrary nodes (including non-convex and degenerate elements). Numerical experiments are carried out to confirm the convergence analysis and illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper investigates a novel C 0 nonconforming virtual element method (VEM) for solving the Kirchhoff plate obstacle problem, which is described by a fourth-order variational inequality (VI) of the first kind. In our study, we distinguish our approach by introducing new internal degrees of freedom to the traditional lowest-order C 0 nonconforming VEM, which originally lacked such degrees. This addition not only facilitates error estimation but also enhances its intuitiveness. Importantly, our novel C 0 nonconforming VEM naturally satisfies the constraints of the obstacle problem. We then establish an a priori error estimate for our novel C 0 nonconforming VEM, with the result indicating that the lowest order of our method achieves optimal convergence. Finally, we present a numerical example to validate the theoretical result. [ABSTRACT FROM AUTHOR]

In this paper a two grid algorithm for quasilinear elliptic problem based on virtual element method (VEM) discretization is proposed. With this new algorithm the solution of a quasilinear elliptic problem on a fine grid is reduced to the solution of a quasilinear elliptic problem on a much coarser grid, and the solution of a linear system on the fine grid. A priori error estimate in H1 norm is derived. Numerical experiments are carried out to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, a fractional optimal control problem with fractional Laplacian and integral state constraint is investigated. The first-order optimality system and regularity of the solutions to optimal control problem are derived. Piecewise linear finite element method combined with variational discretization approach is used to approximate the optimal control problem. A priori error estimates for finite element approximation of the state, adjoint state and control variable are derived. Numerical examples are given to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we develop a two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes. The L1 graded mesh scheme is considered in the time direction, and the VEM is used to approximate spatial direction. The two-grid virtual element algorithm reduces the solution of the nonlinear time fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithm not only saves total computational cost, but also maintains the optimal accuracy. Optimal L 2 error estimates are analysed in detail for both the VEM scheme and the corresponding two-grid VEM scheme. Finally, numerical experiments presented confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper investigates a novel C0 nonconforming virtual element method (VEM) for solving the Kirchhoff plate obstacle problem, which is described by a fourth-order variational inequality (VI) of the first kind. In our study, we distinguish our approach by introducing new internal degrees of freedom to the traditional lowest-order C0 nonconforming VEM, which originally lacked such degrees. This addition not only facilitates error estimation but also enhances its intuitiveness. Importantly, our novel C0 nonconforming VEM naturally satisfies the constraints of the obstacle problem. We then establish an a priori error estimate for our novel C0 nonconforming VEM, with the result indicating that the lowest order of our method achieves optimal convergence. Finally, we present a numerical example to validate the theoretical result.

In this paper, we present a class of discontinuous Galerkin finite element methods for advection-diffusion-reaction problems and establish a priori error estimates when the solution is only in H 1 + s (Ω) with s ∈ (0 , 1 / 2 ]. [ABSTRACT FROM AUTHOR]

In this work, a fast second-order finite volume element (FVE) algorithm is proposed to solve the nonlinear time-fractional coupled diffusion model based on the time two-mesh (TT-M) computing method. In this algorithm, the integer and Riemann-Liouville fractional derivatives are approximated by the second-order backward difference formula and the WSGD formula respectively, the time interval is divided into coarse and fine meshes, then the three steps TT-M FVE algorithm is constructed by using the interpolation operator. The existence and uniqueness for the TT-M FVE algorithm are analyzed in detail, the asymptotically optimal a priori error estimates for variables u and v in the discrete L ∞ (L 2 (Ω)) and L2(H1(Ω) norms are obtained. It is shown that when time coarse and fine mesh sizes satisfy τ c = O (τ f 1 / 2) , the fast algorithm can achieve the same accuracy as the FVE algorithm, and reduce more computational cost. Finally, some numerical results are given to demonstrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a discontinuous Galerkin finite element method of Nitsche's version for the Steklov eigenvalue problem in linear elasticity is presented. The a priori error estimates are analyzed under a low regularity condition, and the robustness with respect to nearly incompressible materials (locking-free) is proven. Furthermore, some numerical experiments are reported to show the effectiveness and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, we study mixed virtual element methods for a distributed optimal control problem governed by a fourth order partial differential equation. By introducing an auxiliary variable, the fourth order equation can be transformed into systems of second order equations. A mixed virtual element discrete scheme for the optimal control problem is established. Moreover, a priori error estimates for auxiliary, state, adjoint state and control variable in and norms are derived. Finally, the theoretical finding is verified by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we study the virtual element discretization of an elliptic optimal control problem with pointwise Neumann boundary control constraint. We construct a virtual element discrete scheme using virtual element discretzation of the state equation and variational discretization of the control variable. A priori error estimates for the state, adjoint state and control variable with respect to H 1 and L 2 norm are derived. Finally, numerical examples are presented to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, time fractional reaction–diffusion equations with the Caputo fractional derivative are solved by using the classical L 1 -formula and the finite volume element (FVE) methods on triangular grids. The existence and uniqueness for the fully discrete FVE scheme are given. The stability results and optimal a priori error estimate in L 2 (Ω) -norm are derived, but it is difficult to obtain the corresponding results in H 1 (Ω) -norm, so another analysis technique is introduced and used to achieve our goal. Finally, some numerical results are given to verify the feasibility and effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, we focus on constructing and analyzing a superconvergent ultra-weak local discontinuous Galerkin (UWLDG) method for the one-dimensional nonlinear fourth-order equation of the form − u(4) = f(x, u). We combine the advantages of the local discontinuous Galerkin (LDG) method and the ultra-weak discontinuous Galerkin (UWDG) method. First, we rewrite the fourth-order equation into a second-order system, then we apply the UWDG method to the system. Optimal error estimates for the solution and its second derivative in the L2-norm are established on regular meshes. More precisely, we use special projections to prove optimal error estimates with order p + 1 in the L2-norm for the solution and for the auxiliary variable approximating the second derivative of the solution, when piecewise polynomials of degree at most p and mesh size h are used. We then show that the UWLDG solutions are superconvergent with order p + 2 toward special projections of the exact solutions. Our proofs are valid for arbitrary regular meshes using Pp polynomials with p ≥ 2. Finally, various numerical examples are presented to demonstrate the accuracy and capability of our method. [ABSTRACT FROM AUTHOR]

In this paper a two grid algorithm for semilinear elliptic problem based on virtual element method (VEM) discretization is proposed. With this new algorithm the solution of a semilinear elliptic problem on a fine grid is reduced to the solution of a semilinear elliptic problem on a much coarser grid, and the solution of a linear system on the fine grid. A priori error estimates in H 1 and L 2 norms are derived. Numerical experiments are carried out to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes a superconvergent ultra-weak discontinuous Galerkin (UWDG) finite element method for nonlinear second-order two-point boundary-value problems. We first derive optimal L 2 -error estimates of the scheme. The order of convergence is proved to be p + 1 in the L 2 -norm, when piecewise polynomials of degree p ≥ 2 are used. Moreover, we prove that the UWDG solution is superconvergent with order p + 2 for p = 2 and p + 3 for p ≥ 3 towards a special projection of the exact solution. Finally, we prove that the UWDG solution and its derivative are superconvergent at nodes with order 2p. Our proofs are valid for arbitrary regular meshes using piecewise polynomials with degree p ≥ 2 . Numerical experiments are presented to confirm the sharpness of all the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the first-order and second-order perturbation approximation schemes for an optimal control problem governed by parabolic PDEs with small uncertainties. First, we propose the finite dimensional noise assumption for the random coefficient, and expand the state and co-state variables up to a certain order with respect to a small parameter by using the perturbation technique. Then, we insert all the expansions into the known deterministic parametric optimality system, and obtain the first-order and second-order optimality systems. These two systems are discretized by the finite element method in space and the backward Euler method in time to establish the first-order and second-order approximate schemes. Further, a priori error estimates are derived for the state, co-state and control variables of the first-order and second-order approximation, respectively. Finally, some numerical experiments are presented to verify the theoretical results. • Use the perturbation technique to expand the state and co-state variables. • Obtain the deterministic first-order and second-order optimality systems. • Establish the first-order and second-order approximate schemes. • Derive a priori error estimates for the state, co-state and control variables. [ABSTRACT FROM AUTHOR]

Standard discontinuous Galerkin methods, based on piecewise polynomials of degree q = 0 , 1 , are considered for temporal semi-discretization for second-order hyperbolic equations. The main goal of this paper is to present a simple and straightforward a priori error analysis of optimal order with minimal regularity requirement on the solution. Uniform norm in time error estimates are also proved. To this end, energy identities and stability estimates of the discrete problem are proved for a slightly more general problem. These are used to prove optimal order a priori error estimates with minimal regularity requirement on the solution. The combination with the classic continuous Galerkin finite element discretization in space variable is used to formulate a full-discrete scheme. The a priori error analysis is presented. Numerical experiments are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the hybridized, embedded-hybridized and embedded discontinuous Galerkin methods for the Stokes equations with Dirac measures. The velocity, the velocity traces and the pressure traces are approximated by polynomials of degree k ≥ 1 , and the pressure is discretized by polynomials of degree k - 1 . An attractive property, named divergence-free, is satisfied by the discrete velocity field. Moreover, the discrete velocity fields derived by hybridized and embedded-hybridized discontinuous Galerkin methods are H(div)-conforming. Using duality argument and Oswald interpolation, a priori and a posteriori error estimates are obtained for the velocity in L 2 -norm. In addition, a posteriori error estimates for the velocity in W 1 , q -seminorm and the pressure in L q -norm are also derived. Finally, numerical examples are provided to validate the theoretical analysis and show the performance of the obtained a posteriori error estimators. [ABSTRACT FROM AUTHOR]

In this paper, we develop and analyze an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order initial-value problems for ordinary differential equations of the form u ″ = f (x , u). Our main concern is to study the convergence and superconvergence properties of the proposed scheme. With a suitable choice of the numerical fluxes, we prove the optimal error estimates with order p + 1 in the L 2 -norm for the solution, when piecewise polynomials of degree at most p are used. We use these results to prove that the UWDG solution is superconvergent with order p + 2 for p = 2 and p + 3 for p ≥ 3 towards a special projection of the exact solution. We further prove that the p -degree UWDG solution and its derivative are (h 2 p) superconvergent at the end of each step. Our proofs are valid for arbitrary regular meshes using piecewise polynomials with degree p ≥ 2. Finally, numerical experiments are provided to verify that all theoretical findings are sharp. The main advantage of our method over the standard DG method for systems of first-order equations is that the UWDG method can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system, which reduces memory and computational costs. [ABSTRACT FROM AUTHOR]

Abstract As an extension of the finite element method, the virtual element method (VEM) can handle very general polygonal meshes, making it very suitable for non-matching meshes. In (Wriggers et al. in Comput. Mech. 58:1039–1050, 2016), the lowest-order virtual element method was applied to solve the contact problem of two elastic bodies on non-matching meshes. The numerical experiments showed the robustness and accuracy of the virtual element scheme. In this paper, we establish a priori error estimate of the virtual element method for the contact problem and prove that the lowest-order VEM achieves linear convergence order, which is optimal.

Abstract We present a new procedure for the numerical study of the wave equation. We use the spectral discretization method associated with the Euler scheme for spatial and temporal discretization. A detailed numerical analysis leads to an a priori error estimate. We confirm the high precision of the method presented by a numerical study.

In this paper we investigate virtual element discretization of optimal control problem governed by Stokes equations. Based on the strategy of first-discretize-then-optimize we build up the virtual element discrete scheme and derive the discrete first order optimality system. A priori error estimates for the state, adjoint state and control variables in L 2 and H 1 norms are derived. Numerical experiments are presented to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

A hp -finite element discretization of a Bingham type variational inequality of the second kind is being analyzed. We prove convergence, including guaranteed convergence rates in the mesh size h and polynomial degree p of the discrete FE-solution. The friction functional may be regularized in which case we also prove convergence and convergence rates in the regularization parameter ϵ. Moreover, we derive two families of reliable a posteriori error estimators which are applicable to any "approximation" of the exact solution and not only to the FE-solution and can therefore be coupled with an iterative solver. The exact computable member of each family of a posteriori error estimators with minimal value is proven to satisfy an efficiency estimate. Numerical results underline the theoretical finding. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a spherically symmetric inverse heat conduction problem, which determines the internal surface temperature distribution of the hollow sphere from measured data at the fixed location inside it. This problem is ill-posed, and a conditional stability result of its solution is given. A modified quasi-boundary value method is proposed to solve the ill-posed problem. Two H o ¨ lder-type error estimates between the approximation solution and its exact solution are obtained under an a priori and an a posteriori regularization parameter selection rule, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonconforming Crouzeix–Raviart (CR) element approximation for the fluid–solid vibration problem. We present the a priori error estimate and the a posteriori error estimate, and we prove the reliability and efficiency of the a posteriori error indicator up to higher order terms. Moreover, we carry out the numerical experiments in adaptive fashion which together with theoretical analysis show that the CR element eigenvalues approximate the exact ones from below. [ABSTRACT FROM AUTHOR]

We develop and analyze a framework for consistent QM/MM (quantum/classic) hybrid models of crystalline defects, which admits general atomistic interactions including traditional off-the-shell interatomic potentials as well as state of art "machine-learned interatomic potentials." We (i) establish an a priori error estimate for the QM/MM approximations in terms of matching conditions between the MM and QM models, and (ii) demonstrate how to use these matching conditions to construct practical machine learned MM potentials specifically for QM/MM simulations. [ABSTRACT FROM AUTHOR]

In this article a spectral Galerkin approximation of fractional advection diffusion optimal control problem with state constraint in L2 norm is discussed. The continuous first order optimality condition is derived. Weighted Jacobi polynomials are used to approximate the state and adjoint state. A priori error estimates for state, adjoint state, and control variables in Hα/2 and weighted L2 norms are derived. Numerical examples are presented to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

In this work we analyze how quadrature rules of different precisions and piecewise polynomial test functions of different degrees affect the convergence rate of Variational Physics Informed Neural Networks (VPINN) with respect to mesh refinement, while solving elliptic boundary-value problems. Using a Petrov-Galerkin framework relying on an inf-sup condition, we derive an a priori error estimate in the energy norm between the exact solution and a suitable high-order piecewise interpolant of a computed neural network. Numerical experiments confirm the theoretical predictions and highlight the importance of the inf-sup condition. Our results suggest, somehow counterintuitively, that for smooth solutions the best strategy to achieve a high decay rate of the error consists in choosing test functions of the lowest polynomial degree, while using quadrature formulas of suitably high precision. [ABSTRACT FROM AUTHOR]

In this paper we investigate the finite element approximation of time fractional optimal control problem with integral state constraint. A space-time finite element scheme for the control problem is developed with piecewise constant time discretization and piecewise linear spatial discretization for the state equation. A priori error estimate for the space-time discrete scheme is derived. Projected gradient algorithm is used to solve the discrete optimal control problem. Numerical experiments are carried out to illustrate the theoretical findings.

In this paper, a virtual element method (VEM) discretization of elliptic optimal control problem with pointwise control constraint is investigated. Virtual element discrete scheme is constructed based on virtual element approximation of the state equation and variational discretization of the control variable. A priori error estimates for state, adjoint state and control variable in H1 and L2 norms are derived. Due to the attractive flexibility of VEM in dealing with mesh refinement we also derive a posteriori error estimates for the optimal control problem, which are used to guide the mesh refinement in the adaptive VEM algorithm. Numerical experiments are carried out to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we present a two-grid scheme of expanded mixed finite element method combined with two second-order time discretization schemes for semilinear parabolic integro-differential equations and give a detailed convergence analysis. On the coarse grid space, we first use the Crank–Nicolson scheme to solve the original nonlinear problem at the first time step, then we utilize the Leap-Frog scheme at the rest time levels. Next, we make use of the known coarse mesh solution and Taylor expansion to infer the solution on the fine mesh space. Thus, we only need to solve the nonlinear problem once on the coarse grid space of the two-grid scheme. Finally, a numerical example is presented to verify the effectiveness of the proposed two-grid scheme. [ABSTRACT FROM AUTHOR]

As an extension of the finite element method, the virtual element method (VEM) can handle very general polygonal meshes, making it very suitable for non-matching meshes. In (Wriggers et al. in Comput. Mech. 58:1039–1050, 2016), the lowest-order virtual element method was applied to solve the contact problem of two elastic bodies on non-matching meshes. The numerical experiments showed the robustness and accuracy of the virtual element scheme. In this paper, we establish a priori error estimate of the virtual element method for the contact problem and prove that the lowest-order VEM achieves linear convergence order, which is optimal. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the first-order and second-order perturbation approximation schemes for an optimal control problem governed by elliptic PDEs with small uncertainties. The optimal control minimizes the expectation of a cost functional with a deterministic constrained control. First, using a perturbation method, we expand the state and co-state variables up to a certain order with respect to a parameter that controls the magnitude of uncertainty in the input. Then we take the expansions into the known deterministic parametric optimality system to derive the first-order and second-order optimality systems which are both deterministic problems. After that, the two systems are discretized by finite element method directly. The strong and weak error estimates are derived for the state, co-state and control variables, respectively. We finally illustrate the theoretical results by two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the discontinuous Galerkin finite element method for the Steklov eigenvalue problem arising in inverse scattering. We present a complete error estimates including the a refined priori error estimate and the a posteriori error estimate, and prove the reliability and efficiency of the a posteriori error estimators for eigenfunctions up to higher order terms, and we also analyze the reliability of estimators for eigenvalues. Moreover, we carry out the numerical experiments in adaptive fashion which together with theoretical analysis show that our method reach the optimal convergence order. [ABSTRACT FROM AUTHOR]

We present a new procedure for the numerical study of the wave equation. We use the spectral discretization method associated with the Euler scheme for spatial and temporal discretization. A detailed numerical analysis leads to an a priori error estimate. We confirm the high precision of the method presented by a numerical study. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac measures. Under assumption that the domain is convex and the mesh is quasi-uniform, a priori error estimate for the error in L2-norm is proved. By duality argument and Oswald interpolation, a posteriori error estimates for the errors in L2-norm and W1,p-seminorm are also obtained. Finally, numerical examples are provided to validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

We consider a moving boundary problem with kinetic condition that describes the diffusion of solvent into rubber and study semi-discrete finite element approximations of the corresponding weak solutions. We report on both a priori and a posteriori error estimates for the mass concentration of the diffusants, and respectively, for the a priori unknown position of the moving boundary. Our working techniques include integral and energy-based estimates for a nonlinear parabolic problem posed in a transformed fixed domain combined with a suitable use of the interpolation-trace inequality to handle the interface terms. Numerical illustrations of our FEM approximations are within the experimental range and show good agreement with our theoretical investigation. This work is a preliminary investigation necessary before extending the current moving boundary modeling to account explicitly for the mechanics of hyperelastic rods to capture a directional swelling of the underlying elastomer. [ABSTRACT FROM AUTHOR]

We derive results on equivalence of piecewise polynomial approximations of a given function in the Sobolev space H (curl) . We namely show that the global-best approximation of a given H (curl) function in a H (curl) -conforming piecewise polynomial space imposing the continuity of the tangential trace can be bounded above and below by the Hilbertian sum of the respective local approximations from the elementwise spaces without any inter-element continuity requirement. In other words, the approximation of a H (curl) function by tangential-trace-continuous and discontinuous piecewise polynomials has comparable precision. We consider approximations of the curl of the target function in the L 2 -norm, as well as approximations of the target function in the L 2 -norm with a constraint on the curl; in the latter case, the constraint is removed in the local approximations. These best-approximation localizations hold under the minimal H (curl) regularity, on arbitrary shape-regular tetrahedral meshes, and include imposition of conditions on a part of the boundary. They extend to the H (curl) context some recent results from the H 1 and H (div) spaces and have direct applications to a priori and a posteriori error analysis of numerical discretizations related to the H (curl) space, namely Maxwell’s equations. [ABSTRACT FROM AUTHOR]

A control problem for a linearized time-discrete regularized fracture propagation process is considered. The discretization of the problem is done using a conforming finite element method. In contrast to many works on discretization of PDE constrained optimization problems, the particular setting has to cope with the fact that the linearized fracture equation is not necessarily coercive. A quasi-best approximation result will be shown in the case of an invertible, though not necessarily coercive, linearized fracture equation. Based on this a priori error estimates for the control, state, and adjoint variables will be derived. [ABSTRACT FROM AUTHOR]

In this paper, we propose L 2 (J ; H 0 1 (Ω)) and L 2 (J ; L 2 (Ω)) norm error estimates that provide the explicit values of the error constants for the semi-discrete Galerkin approximation of the linear heat equation. The derivation of these error estimates shows the convergence of the approximation to the weak solution of the linear heat equation. Furthermore, explicit values of the error constants for these estimates play an important role in the computer-assisted existential proofs of solutions to semi-linear parabolic partial differential equations. In particular, the constants provided in this paper are better than the existing constants and, in a sense, the best possible. [ABSTRACT FROM AUTHOR]

In this paper spectral Galerkin approximation of optimal control problem governed by fractional advection diffusion reaction equation with integral state constraint is investigated. First order optimal condition of the control problem is discussed. Weighted Jacobi polynomials are used to approximate the state and adjoint state. A priori error estimates for control, state, adjoint state and Lagrangian multiplier are derived. Numerical experiment is carried out to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, a second order Gauge Uzawa scheme for the governing equations of an incompressible micropolar fluid flow is introduced and analyzed. The derivation of this scheme is obtained using the second order backward difference approximation. Later, the unconditional stability of the GUM scheme for the micropolar equations will be shown. Then, an a priori error estimate is established to prove the convergence. Finally, we present some numerical simulations that confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a spherically symmetric inverse heat conduction problem, which determines the internal surface temperature distribution of the hollow sphere from measured data at the fixed location inside it. This problem is ill-posed, and a conditional stability result of its solution is given. A modified quasi-boundary value method is proposed to solve the ill-posed problem. Two Ho¨lder-type error estimates between the approximation solution and its exact solution are obtained under an a priori and an a posteriori regularization parameter selection rule, respectively.