Problems In Mixed Form Research Articles

Unconditional error analysis of linearized BDF2 mixed virtual element method for semilinear parabolic problems on polygonal meshes

In this paper, we construct, analyze, and numerically validate a class of H(div)-mixed virtual element method for the semilinear parabolic problem in mixed form, in which the parabolic problem is reformulated in terms of the velocity and the pressure of the time-dependent Darcy flow. The Newton linearized method for the nonlinear term is designed to cooperate with the second-order backward differentiation formula of the temporal discretization scheme. This allows each time step to only require the solution of a small and well-structured linear system rather than the solution of a nonlinear system. The linearization improves computational efficiency without decreasing convergence rates. Moreover, the “Fortin” operator and its approximation properties are applied to derive an optimal error estimates O(hk+1+τ2) for the virtual element solution of the velocity and the pressure. Finally, its remarkable performance is illustrated by several numerical examples that also validate the theoretical rates of convergence.

Numerical analysis of a FEM based on a time-primitive of the electric field for solving a nonlinear transient eddy current problem

The aim of this paper is to analyze from a mathematical and a numerical point of view a formulation of a transient eddy current problem in terms of a time-primitive of the electric field in a bounded domain with ferromagnetic conductors. To this aim, we introduce a Lagrange multiplier to impose the free-divergence condition in the isolated domain. Thus, we obtain a nonlinear degenerate parabolic problem in mixed form and prove its well-posedness. Then, we propose a fully-discrete scheme by using an implicit Euler scheme for time-discretization and a finite element method based on edge and nodal elements for the spatial discretization. We prove quasi-optimal error estimates for the approximation and include some numerical examples to confirm the obtained theoretical results.

Formula omitted]-conforming virtual element method for the Laplacian eigenvalue problem in mixed form

In this paper, we propose the Stokes-type virtual element method for the Laplacian eigenvalue problem in mixed form, in which the eigenvalue problem is reformulated in terms of the velocity and the pressure of the Darcy flow. The virtual element method is H1-conforming and divergence-free. By applying these properties, the inf-sup condition and the spectral approximation theory of compact operator, we derive the optimal a priori error estimates. Then we focus on the a posteriori error estimates. Based on the superconvergence analysis, the Helmholtz decomposition and the properties of bubble functions, we prove the upper and lower bounds of the residual-type a posteriori error estimator with respect to the approximation error. Finally some numerical results are given to validate the theoretical analysis and present the good performance of the adaptive algorithm.

Forward modeling of direct-current resistivity data on unstructured grids using an adaptive mimetic finite-difference method

An effective solver for the direct-current (DC) resistivity forward-modeling problem should be capable of accommodating arbitrary electrode layouts, adapting the mesh to topography and complex geologic structures, and adaptively refining the mesh to ensure high solution accuracy. We use the capability of the mimetic finite-difference method in naturally accommodating meshes with nonconforming elements to develop adaptive and parallel cell-based (CB) and vertex-based (VB) mimetic schemes for the forward modeling of DC resistivity data on unstructured meshes. A diffusion problem in mixed form and Poisson’s problem are solved in the CB and VB schemes, respectively. The mesh adaptivity involves an iterative h-refinement conducted by a regular subdivision of the marked elements together with a 2-irregularity condition. Goal-oriented residual- and gradient-based error estimators are used to mark the elements for refinement in the CB and VB schemes, respectively. To evaluate the accuracy and efficiency of the mimetic schemes, we use two benchmark models with analytical solutions and standard linear and quadratic finite-element solutions. The numerical results for apparent resistivity from both mimetic schemes accurately approximate the reference analytical and numerical values. Furthermore, on similar meshes, the VB mimetic scheme is found to be less demanding than the CB method in terms of computational resource requirement. We also develop a grid-based mesh generation technique based on the body-centered cubic lattice to generate high-quality initial meshes. Using benchmark examples, we validate this mesh generation method and demonstrate its versatility by discretizing a model of a thin dike where a standard Delaunay mesh generator fails to generate a mesh. Moreover, we display the effectiveness of the presented mimetic approach and the grid-based mesh generation technique using a realistic example with topography and parallelization over a large number of sources.

A modified weak Galerkin finite element method for the linear elasticity problem in mixed form

In this paper, we present and analyze the modified weak Galerkin (MWG) finite element method to the linear elasticity problem in a mixed form. Comparing with weak Galerkin (WG) finite element method, the structure of modified weak Galerkin (MWG) finite element method is especial. The main idea of MWG finite element method for elasticity problem is to express to boundary stress tensor defined on the partition boundary in terms of the average of the interior stress tensor defined inside partition elements. This can substantially reduce the degrees of freedom of the linear system. The well-posedness of the solution and inf–sup condition are investigated in this paper. We obtain the optimal error estimates in both energy norm and L2 norm. Finally, some numerical experiments are exhibited to illustrate the efficiency and accuracy of the modified weak Galerkin finite element method.

Virtual Element Methods for three-dimensional Hellinger-Reissner elastostatic problems

Abstract This note aims at illustrating the application of the Virtual Element Method to elasticity problems in mixed form, following the Hellinger-Reissner variational principle. In order to highlight the potential and the flexibility of our approach, we focus on a three-dimensional low-order Virtual Element scheme, but similar considerations apply to two-dimensional and higher-order methods.

Language Decoding From Visual Neural Activity Evoked by Natural Images

The paper is concerned with the construction and convergence analysis of the simplest C0 virtual element method for the Cahn–Hilliard problem in mixed form. This virtual element method leads to a low requirement on regularity and a treatment of general polygonal elements, including non-convex and degenerate elements. Moreover, by introducing two elliptic operators, we prove the L2-error estimate for concentration ϕ in the semidiscrete scheme. Furthermore, numerical results of the full discrete scheme are presented.

A priori error analysis for a mixed VEM discretization of the spectral problem for the Laplacian operator

The aim of the present work is to derive error estimates for the Laplace eigenvalue problem in mixed form, implementing a virtual element method. With the aid of the theory for non-compact operators, we prove that the proposed method is spurious free and convergent. Optimal order of convergence for the eigenvalues and eigenfunctions are derived. Finally, we report numerical tests to confirm the theoretical results together with a rigorous computational analysis of the effects of the stabilization parameter, inherent for the virtual element methods, in the computation of the spectrum.

Forward modelling of gravity data on unstructured grids using an adaptive mimetic finite-difference method

One of the techniques to optimize the gradient-based inversion of geophysical gravity data is to deal with the large and dense Jacobian matrices implicitly. This is made possible by solving the normal systems iteratively and to use pseudo-forward problems to calculate the effect of the Jacobian matrices. For this, numerical forward solvers should be used instead of analytical solvers. In this study, we propose mimetic finite-difference schemes for the solution of the forward gravity problem on unstructured grids. While the mimetic finite-difference method shares many traits with the finite-element and finite-volume methods, it has the advantage of naturally accommodating grids with arbitrary polyhedral elements. We use cell-based and vertex-based mimetic finite-difference schemes to solve a diffusion problem in mixed form, and Poisson's problem, respectively. The cell-based scheme is solved for the gravitational potential and attraction at the centroids and the facets of the elements, respectively, and the vertex-based scheme is solved for the potential at the vertices of the elements. We compare these mimetic finite-difference schemes with cell-based and vertex-based finite-volume schemes and a vertex-based finite-element scheme, in terms of accuracy of the potential. We then utilize the versatility of the mimetic finite-difference method, in accommodating arbitrary elements, and develop an adaptive mesh refinement tool. The mesh adaptivity is performed by an iterative h-refinement where goal-oriented error estimates are used to mark the elements for refinement. The marked elements are decomposed into new tetrahedra by regular subdivision. Since arbitrary polyhedra are naturally permitted in the mimetic finite-difference method, the added nodes are not regarded as hanging nodes, and therefore, any modification of the scheme or extra refinement is avoided. This characteristic allows preserving the quality of the initial grid and simplifying the refinement procedure. We demonstrate the practicality of the presented mesh adaptivity for the forward modelling of gravity and gravity gradient data using realistic geological models. We show that the cell-based mimetic scheme is more efficient than the vertex-based scheme in terms of computational resources. We also demonstrate that in the presence of irregular interfaces in the input domain, the regular refinement approach, presented in this study, is more reliable than a standard mesh regeneration technique.

Least‐squares virtual element method for the convection‐diffusion‐reaction problem

AbstractIn this paper, we introduce a least‐squares virtual element method for the convection‐diffusion‐reaction problem in mixed form. We use the (div) virtual element and continuous virtual element to approximate the flux and the primal variables, respectively. The method allows for the use of very general polygonal meshes. Optimal order a priori error estimates are established for the flux and the primal variables in suitable norms. The least‐squares method offers an efficient a posteriori error estimator without extra effort. Moreover, the hanging nodes are naturally treated in the virtual element method, which provides the high flexibility in mesh refinement because the local mesh postprocessing is never required. Both attractive features motivate us to develop the a posteriori error estimate of the method. Numerical experiments are shown to illustrate the accuracy of the theoretical analysis and demonstrate that the adaptive mesh refinement driven by the proposed estimator can efficiently capture the boundary and the interior layers.

An observation on the uniform preconditioners for the mixed Darcy problem

AbstractWhen solving a multiphysics problem one often decomposes a monolithic system into simpler, frequently single‐physics, subproblems. A comprehensive solution strategy may commonly be attempted, then, by means of combining strategies devised for the constituent subproblems. When decomposing the monolithic problem, however, it may be that requiring a particular scaling for one subproblem enforces an undesired scaling on another. In this manuscript we consider the H(div)‐based mixed formulation of the Darcy problem as a single‐physics subproblem; the hydraulic conductivity, K, is considered intrinsic and not subject to any rescaling. Preconditioners for such porous media flow problems in mixed form are frequently based on H(div) preconditioners rather than the pressure Schur complement. We show that when the hydraulic conductivity, K, is small the pressure Schur complement can also be utilized for H(div)‐based preconditioners. The proposed approach employs an operator preconditioning framework to establish a robust, K‐uniform block preconditioner. The mapping property of the continuous operator is a key component in applying the theoretical framework point of view. As such, a main challenge addressed here is establishing a K‐uniform inf‐sup condition with respect to appropriately weighted Hilbert intersection‐ and sum spaces.

A virtual element method for the Laplacian eigenvalue problem in mixed form

In this paper, the virtual element method for the approximation of Laplacian eigenvalue problem in mixed form is studied. We show that the discrete form satisfies the hypotheses required by the Brezzi-Babǔska theory. Under some assumptions on polygonal meshes, we employ the spectral theory of compact operators to prove the spectral approximation and the optimal order for the eigenvalues. Finally, some numerical results show that numerical eigenvalues obtained by the proposed numerical scheme can achieve the optimal convergence order.

Mixed‐primal Discontinuous Galerkin approximation of flows in fractured porous media on polygonal and polyhedral grids

AbstractWe propose a formulation based on discontinuous Galerkin methods on polygonal/polyhedral grids for the simulation of flows in fractured porous media. We adopt a model for single‐phase flows where the fracture is modelled as a (d − 1)‐dimensional interface in a d‐dimensional bulk domain and the flow is governed by the Darcy's law in both the bulk and the fracture. The two problems are then coupled through physically consistent conditions. We focus on the numerical approximation of the coupled bulk‐fracture problem, discretizing the bulk problem in mixed form and the fracture problem in primal form. We present an priori h‐ and p‐version error estimate in a suitable (mesh‐dependent) energy norm and numerical tests assessing it.

The High-Order Mixed Mimetic Finite Difference Method for Time-Dependent Diffusion Problems

We propose an arbitrary-order accurate mimetic finite difference (MFD) method for the approximation of time-dependent diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. The method of lines (MOL) is used to combine spatial and temporal discretizations. The spatial scheme requires the definition of a high-order approximation of the divergence and gradient operators and the two inner products for the discrete analogs of fluxes and scalar unknowns. The discrete divergence and gradient operators are built according to a discrete duality relation. The inner product for the flux grid functions is built by explicitly imposing the conditions of consistency and stability. The family of semi-discrete mimetic methods is proved theoretically to be energy-stable as the corresponding continuous problem. Then, a full discretization is derived by combining via MOL the MFD method of order k with time marching schemes from the backward differentiation formula of order $$k+2$$ . Optimal order of accuracy is demonstrated for the scalar variable and verified numerically by solving the time-dependent diffusion problems with a variable diffusion tensor for k from 0 to 3 on three different unstructured mesh families.

Hybridized weak Galerkin finite element method for linear elasticity problem in mixed form

A hybridization technique is applied to the weak Galerkin finite element method (WGFEM) for solving the linear elasticity problem in mixed form. An auxiliary function, the Lagrange multiplier defined on the boundary of elements, is introduced in this method. Consequently, the computational costs are much lower than the standard WGFEM. Optimal order error estimates are presented for the approximation scheme. Numerical results are provided to verify the theoretical results.

A virtual element method for the Cahn–Hilliard problem in mixed form

The paper is concerned with the construction and convergence analysis of the simplest C0 virtual element method for the Cahn–Hilliard problem in mixed form. This virtual element method leads to a low requirement on regularity and a treatment of general polygonal elements, including non-convex and degenerate elements. Moreover, by introducing two elliptic operators, we prove the L2-error estimate for concentration ϕ in the semidiscrete scheme. Furthermore, numerical results of the full discrete scheme are presented.

A Weak Galerkin Finite Element Method for the Linear Elasticity Problem in Mixed Form

A Weak Galerkin Finite Element Method for the Linear Elasticity Problem in Mixed Form

Incomplete factorization by local exact factorization (ILUE)

This study proposes a new preconditioning strategy for symmetric positive (semi-)definite SP(S)D matrices referred to as incomplete factorization by local exact factorization (ILUE). The investigated technique is based on exact LU decomposition of small-sized local matrices associated with a splitting of the domain into overlapping or non-overlapping subdomains. The ILUE preconditioner is defined and its relative condition number estimated. Numerical tests on linear systems arising from the finite element (FE) discretization of a second order elliptic boundary value problem in mixed form demonstrate the advantage of the new algorithm, even for problems with highly oscillatory permeability coefficients, against the classical ILU(p) and ILUT(τ) incomplete factorization preconditioners.

Decoupling mixed finite elements on hierarchical triangular grids for parabolic problems

In this paper, we propose a numerical method for the solution of time-dependent flow problems in mixed form. Such problems can be efficiently approximated on hierarchical grids, obtained from an unstructured coarse triangulation by using a regular refinement process inside each of the initial coarse elements. If these elements are considered as subdomains, we can formulate a non-overlapping domain decomposition method based on the lowest-order Raviart–Thomas elements, properly enhanced with Lagrange multipliers on the boundaries of each subdomain (excluding the Dirichlet edges). A suitable choice of mixed finite element spaces and quadrature rules yields a cell-centered scheme for the pressures with a local 10-point stencil. The resulting system of differential-algebraic equations is integrated in time by the Crank–Nicolson method, which is known to be a stiffly accurate scheme. As a result, we obtain independent subdomain linear systems that can be solved in parallel. The behavior of the algorithm is illustrated on a variety of numerical experiments.

The arbitrary order mixed mimetic finite difference method for the diffusion equation

We propose an arbitrary-order accurate mimetic finite difference (MFD) method for the approximation of diffusion problems in mixed form on unstructured polygonal and polyhedral meshes. As usual in the mimetic numerical technology, the method satisfies local consistency and stability conditions, which determines the accuracy and the well-posedness of the resulting approximation. The method also requires the definition of a high-order discrete divergence operator that is the discrete analog of the divergence operator and is acting on the degrees of freedom. The new family of mimetic methods is proved theoretically to be convergent and optimal error estimates for flux and scalar variable are derived from the convergence analysis. A numerical experiment confirms the high-order accuracy of the method in solving diffusion problems with variable diffusion tensor. It is worth mentioning that the approximation of the scalar variable presents a superconvergence effect.