A New Discontinuous Finite Volume Method for Elliptic Problems

A finite-element discretization, independent of the location of the interface, is proposed and analysed for linear elliptic and parabolic interface problems. We establish error estimates of optimal order in the H 1 -norm and almost optimal order in the L 2 -norm for elliptic interface problems. An extension to parabolic interface problems is also discussed and an optimal error estimate in the L 2 (0, T; H1 (Ω))-norm and an almost optimal order estimate in the L 2 (0, T; L 2 (Ω))-norrn are derived for the spatially discrete scheme. A fully discrete scheme based on the backward Euler method is analysed and an optimal order error estimate in the L 2 (0, T; H I (Ω))-norm is derived. The interfaces are assumed to be of arbitrary shape and smooth for our purpose.

In this paper, we construct and analyze a nonconforming finite volume method (FVM) for solving the elliptic boundary value problems on quadrilateral meshes: the hybrid Wilson FVM. Under the mesh assumption that the underlying mesh is an h2-parallelogram mesh, we show that the scheme possesses first order in the mesh-dependent H1-norm and second order in the L2-norm error estimates, the same optimal convergence orders as those of the corresponding Wilson finite element method (FEM). Numerical results are presented to demonstrate the theoretical results on the convergence order of the method.

A dynamic frictionless contact problem between an elasto-visco-plastic body and a deformable obstacle is numerically studied in this paper. The contact is modeled using the normal compliance contact condition involving adhesion effect of contact surfaces and the thermo-elasto-visco-plastic constitutive law is employed to simulate the thermal effects. The adhesion of the contact surfaces is considered and is modeled with a surface variable, the bonding field, whose evolution is described by a first order differential equation. We provide variational formulation which can be put into a general model of system defined by a second order evolution equality, a first order evolution equation and a first order differential equation. The existence and the uniqueness of the weak solution were established. The proofs are based on evolution equations with monotone operators, differential equations and fixed point theorem. The applied approach leads to a system involving variational equalities and we focuse on the study of fully discrete approximation scheme where both the spatial and temporal variables are discretized. The finite difference method and finite element method are popular numerical methods to discretize problems involving ordinary and partial differential equations. In this work, we will use the first-order finite differences to approximate time derivatives. Moreover, our discussion will be limited to the case of uniform partitions of the time interval and we will use the finite elements for spatial discretizations of variational equalities. For a numerical scheme, we prove the existence and uniqueness of the solutions, and derive optimal order error estimates under certain regularity assumption on the solution of the continuous problem. The additional assumptions on the solution regularity are needed for optimal order error estimates and may not always be valid. Nevertheless, weaker solution regularity assumptions can be used to derive error estimates with lower convergence orders.

In this paper, we establish a novel reduced-order weak Galerkin (ROWG) finite element method for solving parabolic equation with nonlinear compression coefficient. We first present the classical weak Galerkin finite element discretization scheme and derive the optimal error estimates. Then we apply a proper orthogonal decomposition (POD) technique to develop the ROWG method, which can effectively reduce degrees of freedom and CPU time. The optimal order error estimates are also derived, and the algorithm flow is provided. Finally, some numerical experiments illustrate the performance of the ROWG method. The numerical results show that the proposed ROWG method is efficient for solving nonlinear parabolic equations.

Optimal order error estimates in H 1, for the Q 1 isoparametric interpolation were obtained in Acosta and Duran (SIAM J Numer Anal37, 18–36, 1999) for a very general class of degenerate convex quadrilateral elements. In this work we show that the same conlusions are valid in W 1,p for 1≤ p < 3 and we give a counterexample for the case p ≥ 3, showing that the result cannot be generalized for more regular functions. Despite this fact, we show that optimal order error estimates are valid for any p ≥ 1, keeping the interior angles of the element bounded away from 0 and π, independently of the aspect ratio. We also show that the restriction on the maximum angle is sharp for p ≥ 3.

We prove optimal order error estimates in the H1 norm for the $\Cq_1$ isoparametric interpolation on convex quadrilateral elements under rather weak hypotheses, improving previously known results. Choose one diagonal and divide the element into two triangles. We show that, if the chosen diagonal is the longest one, then the constant in the error estimate depends only on the maximum angle of the two triangles. Otherwise, the constant depends on that maximum angle and on the ratio between the two diagonals. In particular, we obtainthe optimal order error estimate under the maximum angle condition as in the case of triangular elements. Consequently, the error estimate is uniformly valid for a rather general class of degenerate quadrilaterals.

A second-order face-centred finite volume method for elliptic problems

Recently, the $\vec{H}(\operatorname{div})$-conforming finite element families for second order elliptic problems have come more into focus, since due to hybridization and subsequent advances in computational efficiency their use is no longer mainly theoretical. Their property of yielding exactly divergence-free solutions for mixed problems makes them interesting for a variety of applications, including incompressible fluids. In this area, boundary and interior layers are present, which demand the use of anisotropic elements. While for the Raviart-Thomas interpolation of any order on anisotropic tetrahedra optimal error estimates are known, this contribution extends these results to the Brezzi-Douglas-Marini finite elements. Optimal interpolation error estimates are proved under two different regularity conditions on the elements, which both relax the standard minimal angle condition. Additionally a numerical application on the Stokes equations is presented to illustrate the findings.

In this paper we develop and study a new stabilized finite volume method for the two-dimensional Stokes equations. This method is based on a local Gauss integration technique and the conforming elements of the lowest-equal order pair (i.e., the P 1–P 1 pair). After a relationship between this method and a stabilized finite element method is established, an error estimate of optimal order in the H 1-norm for velocity and an estimate in the L 2-norm for pressure are obtained. An optimal error estimate in the L 2-norm for the velocity is derived under an additional assumption on the body force.

Quasi-Wilson nonconforming element approximation for nonlinear dual phase lagging heat conduction equations

Numerical analysis of history-dependent hemivariational inequalities and applications to viscoelastic contact problems with normal penetration

A \(C^0\)-weak Galerkin (WG) method is introduced and analyzed in this article for solving the biharmonic equation in 2D and 3D. A discrete weak Laplacian is defined for \(C^0\) functions, which is then used to design the weak Galerkin finite element scheme. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established for the weak Galerkin finite element solution in both a discrete \(H^2\) norm and the standard \(H^1\) and \(L^2\) norms with appropriate regularity assumptions. Numerical results are presented to confirm the theory. As a technical tool, a refined Scott-Zhang interpolation operator is constructed to assist the corresponding error estimates. This refined interpolation preserves the volume mass of order \((k+1-d)\) and the surface mass of order \((k+2-d)\) for the \(P_{k+2}\) finite element functions in \(d\)-dimensional space.

Abstract. The aim of this paper is to analyze a low order finite element method for a stiffened plate. The plate is modeled by Reissner-Mindlin equations and the stiffener by Timoshenko beams equations. The resulting problem is shown to be well posed. In the case of concentric stiffeners it decouples into two problems, one for the in-plane plate deformation and the other for the bending of the plate. The analysis and discretization of the first one is straightforward. The second one is shown to have a solution bounded above and below independently of the thickness of the plate. A discretization based on DL3 finite elements combined with ad-hoc elements for the stiffener is proposed. Optimal order error estimates are proved for displacements, rotations and shear stresses for the plate and the stiffener. Numerical tests are reported in order to assess the performance of the method. These numerical computations demonstrate that the error estimates are independent of the thickness, providing a numerical evidence that the method is locking-free.

A conforming discontinuous Galerkin finite element method for Brinkman equations

Two numerical methods for a Cauchy problem for modified Helmholtz equation