Let $K:=SO\left(2\right)A_1\cup SO\left(2\right)A_2\cdots SO\left(2\right)A_N$ where $A_1,A_2,\cdots ,A_N$ are matrices of non-zero determinant. We establish a sharp relation between the following two minimisation problems in two dimensions. Firstly the $N$-well problem with surface energy. Let $p\in \left[1,2\right]$, $\Omega \subset {ℝ}^2$ be a convex polytopal region. Define$$I_{ϵ}^p\left(u\right)={\int }_{\Omega }{d}^p\left(Du\left(z\right),K\right)+ϵ{\left|D^{2}u\left(z\right)\right|}^2\mathrm{d}{L}^{2}z$$and let $A_F$ denote the subspace of functions in $W^{2,2}\left(\Omega \right)$ that satisfy the affine boundary condition $Du=F$ on $\partial \Omega$ (in the sense of trace), where $F\notin K$. We consider the scaling (with respect to $ϵ$) of$$m_{ϵ}^p:=\underset{u\in A_F}{inf}{I}_{ϵ}^p\left(u\right).$$Secondly the finite element approximation to the $N$-well problem without surface...

Let $K:=SO\left(2\right)A_1\cup SO\left(2\right)A_2\cdots SO\left(2\right)A_N$ where $A_1,A_2,\cdots ,A_N$ are matrices of non-zero determinant. We establish a sharp relation between the following two minimisation problems in two dimensions. Firstly the N-well problem with surface energy. Let $p\in \left[1,2\right]$, $\Omega \subset {ℝ}^2$ be a convex polytopal region. Define $$I_{ϵ}^p\left(u\right)={\int }_{\Omega }{d}^p\left(Du\left(z\right),K\right)+ϵ{\left|D^{2}u\left(z\right)\right|}^2\mathrm{d}{L}^{2}z$$ and let AF denote the subspace of functions in $W^{2,2}\left(\Omega \right)$ that satisfy the affine boundary condition Du=F on $\partial \Omega$ (in the sense of trace), where $F\notin K$. We consider the scaling (with respect to ϵ) of $$m_{ϵ}^p:=\underset{u\in A_F}{inf}{I}_{ϵ}^p\left(u\right).$$ Secondly the finite element approximation to the N-well problem without...

For a general class of atomistic-to-continuum coupling methods, coupling multi-body interatomic potentials with a P1-finite element discretisation of Cauchy–Born nonlinear elasticity, this paper adresses the question whether patch test consistency (or, absence of ghost forces) implies a first-order error estimate. In two dimensions it is shown that this is indeed true under the following additional technical assumptions: (i) an energy consistency condition, (ii) locality of the interface correction,...

For a general class of atomistic-to-continuum coupling methods, coupling multi-body interatomic potentials with a P1-finite element discretisation of Cauchy–Born nonlinear elasticity, this paper adresses the question whether patch test consistency (or, absence of ghost forces) implies a first-order error estimate. In two dimensions it is shown that this is indeed true under the following additional technical assumptions: (i) an energy consistency condition, (ii) locality of the interface correction,...

We estimate the constant in the strengthened Cauchy-Bunyakowski-Schwarz inequality for hierarchical bilinear finite element spaces and elliptic partial differential equations with coefficients corresponding to anisotropy (orthotropy). It is shown that there is a nontrivial universal estimate, which does not depend on anisotropy. Moreover, this estimate is sharp and the same as for hierarchical linear finite element spaces.

We consider a family of shell finite elements with quadratic displacements across the thickness. These elements are very attractive, but compared to standard general shell elements they face another source of numerical locking in addition to shear and membrane locking. This additional locking phenomenon – that we call “pinching locking” – is the subject of this paper and we analyse a numerical strategy designed to overcome this difficulty. Using a model problem in which only this specific source...

We consider a family of shell finite elements with quadratic displacements across the thickness. These elements are very attractive, but compared to standard general shell elements they face another source of numerical locking in addition to shear and membrane locking. This additional locking phenomenon – that we call “pinching locking” – is the subject of this paper and we analyse a numerical strategy designed to overcome this difficulty. Using a model problem in which only this specific source of...

A discrete model of the two-dimensional Signorini problem with Coulomb friction and a coefficient of friction $ℱ$ depending on the spatial variable is analysed. It is shown that a solution exists for any $ℱ$ and is globally unique if $ℱ$ is sufficiently small. The Lipschitz continuity of this unique solution as a function of $ℱ$ as well as a function of the load vector $f$ is obtained. Furthermore, local uniqueness of solutions for arbitrary $ℱ>0$ is studied. The question of existence of locally Lipschitz-continuous...

We consider a viscoelastic solid in Kelvin-Voigt rheology exhibiting also plasticity with hardening and coupled with heat-transfer through dissipative heat production by viscoplastic effects and through thermal expansion and corresponding adiabatic effects. Numerical discretization of the thermodynamically consistent model is proposed by implicit time discretization, suitable regularization, and finite elements in space. Fine a-priori estimates are derived, and convergence is proved by careful successive...

We consider a viscoelastic solid in Kelvin-Voigt rheology exhibiting also plasticity with hardening and coupled with heat-transfer through dissipative heat production by viscoplastic effects and through thermal expansion and corresponding adiabatic effects. Numerical discretization of the thermodynamically consistent model is proposed by implicit time discretization, suitable regularization, and finite elements in space. Fine a-priori estimates are derived, and convergence is proved by careful...

This article presents an idea in the finite element methods (FEMs) for obtaining two-sided bounds of exact eigenvalues. This approach is based on the combination of nonconforming methods giving lower bounds of the eigenvalues and a postprocessing technique using conforming finite elements. Our results hold for the second and fourth-order problems defined on two-dimensional domains. First, we list analytic and experimental results concerning triangular and rectangular nonconforming elements which...