Element Method For Elliptic Problems Research Articles

A recovery-based a posteriori error estimator of the weak Galerkin finite element method for elliptic problems

In this paper, we propose a recovery-type a posteriori error estimator of the weak Galerkin finite element method for the second order elliptic equation. The reliability and efficiency of the estimator are analyzed by a discrete H1-norm of the exact error. The estimator is further used in the adaptive weak Galerkin algorithm on the triangular, quadrilateral and other polygonal meshes. Numerical results are provided to demonstrate the effectiveness of the adaptive mesh refinement guided by this estimator.

A Balanced Oversampling Finite Element Method for Elliptic Problems with Observational Boundary Data

A Balanced Oversampling Finite Element Method for Elliptic Problems with Observational Boundary Data

A combined finite element method for parabolic equations posted in domains with rough boundaries

In this paper, we consider the solution of parabolic equations posted in domains with rough boundaries using the combined finite element procedure proposed in Xu et al. [A combined finite element method for elliptic problems posted in domains with rough boundaries, J. Comput. Appl. Math. 336 (2018), pp. 235–248] in space and backward-Euler scheme in time. We first prove error estimates for elliptic projection operator for both and by the dual argument which depend on the elliptic auxiliary problem, and then energy error estimates of semi-discrete and fully discrete schemes which have convergence rates about and , where s>0, respectively. Numerical results are provided for parabolic equations in domains with non-oscillating or oscillating boundaries to verify the theoretical findings.

Ultraconvergence of the derivative of high‐order finite element method for elliptic problems with constant coefficients

AbstractIn this article, for second order elliptic problems with constant coefficients, the local ultraconvergence of the derivative of finite element method using piecewise polynomials of degrees k (k ≥ 2) is studied by the interpolation postprocessing technique. Under suitable regularity and mesh conditions, we prove that at an interior vertex, which is away from the boundary with a fixed distance, the gradient of the postprecessed finite element solution using piecewise polynomials of degrees k (k ≥ 2) converges to the gradient of the exact solution with order . Numerical experiments are used to illustrate our theoretical findings.

Adaptive Mixed Finite Element Method for Elliptic Problems with Concentrated Source Terms

An adaptive mixed finite element method using the Lagrange multiplier technique is used to solve elliptic problems with delta Dirac source terms. The problem arises in the use of Chow-Anderssen linear functional methodology to recover coefficients locally in parameter estimation of an elliptic equation from a point-wise measurement. In this article, we used a posterior error estimator based on averaging technique as refinement indicators to produce a cycle of mesh adaptation, which is experimentally shown to capture singularity phenomena. Our numerical results showed that the adaptive refinement process successfully refines elements around the center of the source terms. The results also showed that the global error estimation is better than uniform refinement process in terms of computation time.

A new family of discontinuous finite element methods for elliptic problems in the whole space

We investigate a new hybrid method we call the discontinuous Galerkin‐inverted finite element method (DGIFEM) to approximate the solution of elliptic problems in , especially when the growth or the decay of the solution is very slow. On the basis of both the discontinuous Galerkin discretization and the inverted finite element method, the DGIFEM keeps part of the domain bounded and maps the other infinite extent into a bounded region via a suitable polygonal inversion. The numerical solution is then constructed in an appropriate subspace of weighted Sobolev spaces, where the weights allow the control of the growth or the decay of functions at infinity. A careful study of the convergence of the DGIFEM is carried out and shows that the optimal order of convergence can always be reached. Finally, some numerical results are given as illustration of the good performance of the proposed method.

A multiscale virtual element method for elliptic problems in heterogeneous porous media

In this paper, we propose a Multiscale Virtual Element Method (MsVEM) for elliptic problems in heterogeneous porous media. The use of very general coarse grids has advantages in subsurface simulations since they provide flexibility and can render more accurate coarse model. It is often necessary to use an irregular coarse grid when highly heterogeneous reservoirs are discretized via irregular anisotropic fine grids. The proposed MsVEM aims to build a coarse model on a flexible grid by integrating Generalized Multiscale Finite Element Method (GMsFEM) with Virtual Element Method (VEM). Because of the irregularity merit of VEM, MsVEM is desirable to simulate complex real-world problems. In the framework of MsVEM, both coarse grid and fine grid may consist of more general polygons. To build multiscale basis functions on coarse grid, a few local problems are solved on each coarse block to enrich multiscale information and get a snapshot space. To significantly reduce degree of freedoms of the coarse multiscale model, we construct a low dimensional multiscale basis space by using greedy selection procedure and penalized matrix decomposition. The greedy selection procedure is utilized to select optimal multiscale basis functions. The penalized matrix decomposition is employed to improve approximation accuracy and get the sparse basis functions. To show the efficiency and accuracy, we present a few numerical examples for flows in heterogeneous porous media. In particular, MsVEM is applied to the subsurface flow in high-contrast porous media and fractured media.

A multilevel sparse kernel-based stochastic collocation finite element method for elliptic problems with random coefficients

A new stochastic collocation finite element method is proposed for the numerical solution of elliptic boundary value problems (BVP) with random coefficients, assuming that the randomness is well-approximated by a finite number of random variables with given probability distributions. The proposed method consists of a finite element approximation in physical space, along with a stochastic collocation quadrature approach utilizing the recent Multilevel Sparse Kernel-Based Interpolation (MuSIK) technique (Georgoulis et al., 2013). MuSIK is based on a multilevel sparse grid-type algorithm with the basis functions consisting of directionally anisotropic Gaussian radial basis functions (kernels) placed at directionally-uniform grid-points. We prove that MuSIK is interpolatory at these nodes, and, therefore, can be naturally used to define a quadrature scheme. Numerical examples are also presented, assessing the performance of the new algorithm in the context of high-dimensional stochastic collocation finite element methods.

Geometric multigrid algorithms for elliptic interface problems using structured grids

In this work, we develop geometric multigrid algorithms for the immersed finite element methods for elliptic problems with interface (Chou et al. Adv. Comput. Math. 33, 149–168 2010; Kwak and Lee, Int. J. Pure Appl. Math. 104, 471–494 2015; Li et al. Numer. Math. 96, 61–98 2003, 2004; Lin et al. SIAM J. Numer. Anal. 53, 1121–1144 2015). We need to design the transfer operators between levels carefully, since the residuals of finer grid problems do not satisfy the flux condition once projected onto coarser grids. Hence, we have to modify the projected residuals so that the flux conditions are satisfied. Similarly, the correction has to be modified after prolongation. Two algorithms are suggested: one for finite element spaces having vertex degrees of freedom and the other for edge average degrees of freedom. For the second case, we use the idea of conforming subspace correction used for P1 nonconforming case (Lee 1993). Numerical experiments show the optimal scalability in terms of number of arithmetic operations, i.e., $\mathcal {O}(N)$ for $\mathcal {V}$ -cycle and CG algorithms preconditioned with $\mathcal {V}$ -cycle. In $\mathcal {V}$ -cycle, we used only one Gauss-Seidel smoothing. The CPU times are also reported.

An adaptive mesh refinement strategy for finite element solution of the elliptic problem

In this paper we develop an adaptive finite element method for elliptic problems. First, we assume that in each subdomain the norm of the approximation error at the current mesh configuration is bounded by the norm of the approximation error obtained at the previous mesh configuration, for some norm Hs. Then an a-posteriori error estimator is designed based on the approximate solution correction between the solution on the last two mesh configurations. Based on this new error estimator, the element-wise refinement strategy in each subdomain is provided for a given tolerance.A discussion on the choice of the coefficients in the assumption is given for different norm spaces and for different degrees of finite element family. Four 2D numerical benchmark examples of different domains and two 3D numerical benchmark examples are tested to demonstrate the robustness of our method. When possible, our numerical results are also compared to corresponding results from existing methods. All the results show that the proposed method is robust and efficient in terms of the number of degrees of freedom.

The aggregated unfitted finite element method for elliptic problems

Unfitted finite element techniques are valuable tools in different applications where the generation of body-fitted meshes is difficult. However, these techniques are prone to severe ill conditioning problems that obstruct the efficient use of iterative Krylov methods and, in consequence, hindersthe practical usage of unfitted methods for realistic large scale applications. In this work, we present a technique that addresses such conditioning problems by constructing enhanced finite element spaces based on a cell aggregation technique. The presented method, called aggregated unfitted finite element method, is easy to implement, and can be used, in contrast to previous works, in Galerkin approximations of coercive problems with conforming Lagrangian finite element spaces. The mathematical analysis of the method states that the condition number of the resulting linear system matrix scales as in standard finite elements for body-fitted meshes, without being affected by small cut cells, and that the method leads to the optimal finite element convergence order. These theoretical results are confirmed with 2D and 3D numerical experiments.

A combined finite element method for elliptic problems posted in domains with rough boundaries

A combined finite element method is presented in this paper to solve the elliptic problems posted in domains with rough boundaries. Solving these problems numerically is difficult because resolving the boundaries usually requires very fine meshes, while good quality meshes often over-refine unnecessarily the interior of the domain. The basic idea of the proposed method is to use a fine mesh with size h in the vicinity of oscillating boundaries and a coarse mesh with size H ≫ h for other portions of the domain to reduce some unnecessary computational effort. The transmission conditions across the fine-coarse mesh interface are treated by the penalty technique. The key point of the method lies in the new scheme employing a weighted average in the definition of the bilinear form, which avoids the affection of the ratio H ∕ h in the error estimate. We prove a quasi-optimal convergence in terms of elements since there is no whole H 2 regularity in the domain with rough boundaries. Numerical results are provided for elliptic equations in domains with non-oscillating or oscillating boundaries to illustrate the theoretical results.

Two-scale composite finite element method for parabolic problems with smooth and nonsmooth initial data

We consider a new class of finite elements, called composite finite elements, for the discretization of parabolic problems in a two-dimensional convex polygonal domain. More precisely, an effort has been made in this paper to extend two-scale composite finite element method for elliptic problems to parabolic problems. We analyze both semidiscrete and fully discrete composite finite element methods and derive convergence properties in the $$L^\infty (L^2)$$ and $$L^\infty (H^1)$$ norms for smooth data. Moreover, nonsmooth data error estimates in the $$L^\infty (L^2)$$ norm for positive time is established. Compared to the classical finite element method, the composite finite element method can be viewed as a coarse scale generalization which yields a minimal dimension of the approximation space and the asymptotic order of convergence is preserved on coarse scale meshes which do not resolve the boundary. Numerical results are presented to illustrate the theoretical rates of convergence.

Mixed finite element approximations based on 3‐Dhp‐adaptive curved meshes with two types ofH(div)‐conforming spaces

SummaryTwo stable approximation space configurations are treated for the mixed finite element method for elliptic problems based on curved meshes. Their choices are guided by the property that, in the master element, the image of the flux space by the divergence operator coincides with the potential space. By using static condensation, the sizes of global condensed matrices, which are proportional to the dimension of border fluxes, are the same in both configurations. The meshes are composed of different topologies (tetrahedra, hexahedra, or prisms). Simulations using asymptotically affine uniform meshes, exactly fitting a spherical‐like region, and constant polynomial degree distributionk, showL2errors of orderk+1 ork+2 for the potential variable, while keeping orderk+1 for the flux in both configurations. The first case corresponds to RT(k) and BDFM(k+1) spaces for hexahedral and tetrahedral meshes, respectively, but holding for prismatic elements as well. The second case, further incrementing the order of approximation of the potential variable, holds for the three element topologies. The case ofhp‐adaptive meshes is considered for a problem modelling a porous media flow around a cylindrical horizontal well with elliptical drainage area. The effect of parallelism and static condensation in CPU time reduction is illustrated.

Two dimensional mixed finite element approximations for elliptic problems with enhanced accuracy for the potential and flux divergence

The purpose of the present paper is to analyse two new different possibilities of choosing balanced pairs of approximation spaces for dual (flux) and primal (potential) variables, one for triangles and the other one for quadrilateral elements, to be used in discrete versions of the mixed finite element method for elliptic problems. They can be interpreted as enriched versions of BDFMk+1 spaces based on triangles, and of RTk spaces for quadrilateral elements. The new flux approximations are incremented with properly chosen internal shape functions (with vanishing normal components on the edges) of degree k+2, and matching primal functions of degree k+1 (higher than the border fluxes, which are kept of degree k). In all these cases, the divergence of the flux space coincide with the primal approximation space on the master element, producing stable simulations. Using static condensation, the global condensed system to be solved in the enriched cases has same dimension (and structure) of the original ones, which is proportional to the space dimension of the border fluxes for each element geometry. Measuring the errors withL2-norms, the enriched space configurations give higher convergence rate of order k+2 for the primal variable, while keeping the order k+1 for the flux. For affine meshes, the divergence errors have the same improved accuracy rate as for the error in the primal variable. For quadrilateral non-affine meshes, for instance trapezoidal elements, the divergence error has order k+1, one unit more than the order k occurring for RTk spaces on this kind of deformed meshes. This fact also holds for ABFk elements, but for them the potential order of accuracy does not improve, keeping order k+1.

Cut finite element methods for elliptic problems on multipatch parametric surfaces

We develop a finite element method for the Laplace–Beltrami operator on a surface described by a set of patchwise parametrizations. The patches provide a partition of the surface and each patch is the image by a diffeomorphism of a subdomain of the unit square which is bounded by a number of smooth trim curves. A patchwise tensor product mesh is constructed by using a structured mesh in the reference domain. Since the patches are trimmed we obtain cut elements in the vicinity of the interfaces. We discretize the Laplace–Beltrami operator using a cut finite element method that utilizes Nitsche’s method to enforce continuity at the interfaces and a consistent stabilization term to handle the cut elements. Several quantities in the method are conveniently computed in the reference domain where the mappings impose a Riemannian metric. We derive a priori estimates in the energy and L2 norm and also present several numerical examples confirming our theoretical results.

Higher-order finite element methods for elliptic problems with interfaces

We present higher-order piecewise continuous finite element methods for solving a class of interface problems in two dimensions. The method is based on correction terms added to the right-hand side in the standard variational formulation of the problem. We prove optimal error estimates of the methods on general quasi-uniform and shape regular meshes in maximum norms. In addition, we apply the method to a Stokes interface problem, adding correction terms for the velocity and the pressure, obtaining optimal convergence results.

Conforming and nonconforming virtual element methods for elliptic problems

EPSRC (Grant EP/L022745/1) to A.C.; Laboratory Directed Research and Development program (LDRD), US Department of Energy Office of Science, Office of Fusion Energy Sciences, under the auspices of the National Nuclear Security Administration of the US Department of Energy by Los Alamos National Laboratory, operated by Los Alamos National Security LLC under contract DE-AC52- 06NA25396 to G.M.; Ph.D. Studentship from the College of Science and Engineering at the University of Leicester and an EPSRC Doctoral Training Grant to O.S.

Three dimensional hierarchical mixed finite element approximations with enhanced primal variable accuracy

There are different possibilities of choosing balanced pairs of approximation spaces for dual (flux) and primal (pressure) variables to be used in discrete versions of the mixed finite element method for elliptic problems arising in fluid simulations. Three cases shall be studied and compared for discretized three dimensional formulations based on tetrahedral, hexahedral and prismatic meshes. The principle guiding the constructions is the property that the divergence of the dual space and the primal approximation space should coincide, while keeping the same order of accuracy for the flux variable and varying the accuracy order of the primal variable. There is the classic case of BDMk spaces based on tetrahedral meshes and polynomials of total degree k for the dual variable, and k−1 for the primal variable, showing stable simulations with optimal convergence rates of orders k+1 and k, respectively. Another case is related to RTk and BDFMk+1 spaces for hexahedral and tetrahedral meshes, respectively, but holding for prismatic elements as well. It gives identical approximation order k+1 for both primal and dual variables, an improvement in accuracy obtained by increasing the degree of primal functions to k, and by enriching the dual space with some properly chosen internal shape functions of degree k+1, while keeping degree k for the border fluxes. A new type of approximation is proposed by further incrementing the order of some internal flux functions to k+2, and matching primal functions to k+1 (higher than the border fluxes of degree k). Thus, higher convergence rate of order k+2 is obtained for the primal variable. Using static condensation, the global condensed system to be solved in all the cases has same dimension (and structure), which is proportional to the space dimension of the border fluxes for each element geometry. Illustrating results comparing the three different space configurations are presented for simulations based on hierarchical high order shape functions for H(div)-conforming spaces, which are specially constructed for affine tetrahedral, hexahedral and prismatic meshes. Expected convergence rates are obtained for the flux, pressure and divergence variables.

Exponentially accurate nonconforming least-squares spectral element method for elliptic problems on unbounded domain

Non-conforming approximation methods are becoming increasingly popular because of the potential to apply to multi-material and multi-model analysis for both bounded and unbounded domains. In this paper, we present a least-square approximation based method to solve the one or two dimensional elliptic problems on an unbounded domain. The method gives exponential accuracy and shows superior performance when compared to other numerical methods. Differentiability estimates and the main stability estimate theorem, using a non-conforming spectral element method, are also discussed. The exponential convergence rate of the proposed method is also shown through rigorous error estimate and specific numerical examples.