Alternating direction Galerkin method with capacitance matrix for the parabolic problem with natural boundary condition
An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints
We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator...
An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints
We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator...
An adaptive finite element method for solving a double well problem describing crystalline microstructure
The minimization of nonconvex functionals naturally arises in materials sciences where deformation gradients in certain alloys exhibit microstructures. For example, minimizing sequences of the nonconvex Ericksen-James energy can be associated with deformations in martensitic materials that are observed in experiments[2,3]. — From the numerical point of view, classical conforming and nonconforming finite element discretizations have been observed to give minimizers with their quality being highly dependent...
An adaptive finite element method in reconstruction of coefficients in Maxwell's equations from limited observations
We propose an adaptive finite element method for the solution of a coefficient inverse problem of simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions in the Maxwell's system using limited boundary observations of the electric field in 3D. We derive a posteriori error estimates in the Tikhonov functional to be minimized and in the regularized solution of this functional, as well as formulate the corresponding adaptive algorithm. Our numerical experiments...
An Adaptive Newton Algorithm Based on Numerical Inversion: Regularization as Postconditioner.
An additive Schwarz method for mortar Morley finite element discretizations of 4th order elliptic problem in 2D.
An Analysis of a Mixed Finite Element Method for the Navier-Stokes Equations.
An analysis of stability concepts for the Navier-Stokes equations.
An analysis of the boundary layer in the 1D surface Cauchy–Born model
The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error...
An analysis of the boundary layer in the 1D surface Cauchy–Born model∗
The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is 𝒪(1) in the mesh size; however, we are able to identify an alternative “approximation parameter” – the stiffness of the interaction potential – with respect to which the relative error...
An analysis of the cell vertex method
An analysis of the convergence of mixed finite element methods
An Analysis of the p-Version of the Finite Element Method for Nearly Incompressible Materials. Uniformly Valid, Optimal Error Estimates
An analysis technique for stabilized finite element solution of incompressible flows
This paper presents an extension to stabilized methods of the standard technique for the numerical analysis of mixed methods. We prove that the stability of stabilized methods follows from an underlying discrete inf-sup condition, plus a uniform separation property between bubble and velocity finite element spaces. We apply the technique introduced to prove the stability of stabilized spectral element methods so as stabilized solution of the primitive equations of the ocean.
An analysis technique for stabilized finite element solution of incompressible flows
This paper presents an extension to stabilized methods of the standard technique for the numerical analysis of mixed methods. We prove that the stability of stabilized methods follows from an underlying discrete inf-sup condition, plus a uniform separation property between bubble and velocity finite element spaces. We apply the technique introduced to prove the sta bi li ty of stabilized spectral element methods so as stabilized solution of the primitive equations of the ocean.
An analytical and numerical approach to a bilateral contact problem with nonmonotone friction
We consider a mathematical model which describes the contact between a linearly elastic body and an obstacle, the so-called foundation. The process is static and the contact is bilateral, i.e., there is no loss of contact. The friction is modeled with a nonmotonone law. The purpose of this work is to provide an error estimate for the Galerkin method as well as to present and compare two numerical methods for solving the resulting nonsmooth and nonconvex frictional contact problem. The first approach...
An application of the finite element method for solving the system of partial differential equations
An application of the finite element method for solving the system of partial differential equations
An Asymptotic Finite Element Method for Improvement of Solutions of Boundary Layer Problems.