| Poroelasticity model,one of the fluid-structure interaction problems,describes the interaction between the fluid flow and deformation in an elastic porous material.In this model,the motion of fluid is described by Darcy's law,whereas the deformation of the porous medium is governed by linear elasticity.Terzaghi[l]analyzed the one-dimensional(1D)consolidation of a soil column under a constant load.Later,Biot generalized Terzaghi's theory to 3D transient consolidation[2-6].So poroelasticity model is also called Biot's consolidation model.Because of their ubiquity and unique properties,poroelasticity has more and more important applications today in a wide range of science and engineering fields including reservoir engineering[7,9,10],environmental engineering[11,12],biomechanics[13,15]and materials science[16].The analytical solutions of poroelastic problems can be rarely found[17-19]because of the complex coupled nature of the equations.Thus,computational simulations become the key to solving poroelastic problems.The most commonly numerical methods are finite element methods using continuous Galerkin(CG)elements for both displacement and pressure[20],or combine a mixed finite element method for the flow variables and a continuous Galerkin method for displacement[24].However,the numerical solution of standard Galerkin finite element method experiences locking which manifests as pressure oscillations when the constrained specific storage coefficient c0=0,where c0 combines porosity and compressibility.Phillips and Wheeler illustrated that coupled DG/mixed method can eliminate nonphysical pressure oscillations by the numerical experiment[21,25].Then they heuristically examined the reasons for locking in poroelasticity and suggested choosing a approximate space that contains nonconstant divergence-free vectors for the displacement[26].Subsequently,coupled nonconforming and mixed finite element method on triangular meshes and on rectangular meshes are proposed to solve poroelaticity problem[27,50].In view of numerical examples,these methods can also eliminate nonphysical pressure oscillations.In[30,61],they found that the pressure oscillations may be caused by the failure of compatibility for the displacement and pore pressure,rather than elastic locking.Recent years,some new methods dealing with pressure oscillation were proposed for poroelasticity problems,such as WG methods[53,54,58],fully mixed finite element methods[28],stabilized finite element methods[23,29,31-36,52],multipoint flux mixed finite element methods[49],least-squares mixed finite element method[22].Naturally,standard finite difference schemes may suffer the same instable behaviour in pressure approximation.In[62-64],a stabilized space discretization using staggered grids is presented for quasistatic,fully dynamic and double-porosity poroelasticity problem.In that scheme,the x-component of displacement is approximated at the midpoint of horizontal edges of the cell,the y-component of displacement is approximated at the midpoint of vertical edges of the cell,and pressure is approximated at vertex.They proved the error estimations for displacements and pressure in discrete energy norms on uniform grid.All aboved papers only consider the locking phenomenon caused by c0=0.However,standard finite element methods or standard finite difference schemes can suffer from Poisson locking caused by ??? in poroelasticity.But Poisson locking has not received much attention so far in poroelasticity.In[65],they propose a new formulation for Biot's consolidation problem and provide three concrete examples of finite elements subspaces.However,the numerical examples indicates that the lowest-order discretization using[P1]d × P1 x P0 elements does not converge when ???,where d = 2,3.To solve poroelasticity problem,three coupling techniques have been proposed:full coupling,explicit coupling,and iterative coupling.The fully coupled approach solves the coupled problem simultaneously in a time-stepping algorithm.Therefore,it will require special linear solvers to handle the fully coupled system.Explicit coupling is essentially a sequential method.But it is only conditionally stable.The iterative coupling approach solves the problem of flow or the mechanics sequentially at each time step.It can be more stable than explicit coupling,and as accurate as a fully coupled scheme.But it needs iterations until a converged solution within a prescribed numerical tolerance is obtained at each time step.The convergence of the iterative coupling was proved in[55-57].Recently,Nabil Chaabane and Beatrice Riviere proposed a sequential DG[59]and CG[60]method to solve the poroelasticity problem.They partition the coupled problem and solve sub-problems sequentially.Moreover,this approach is fully decoupled and does not require iterations.They proved that the numerical scheme is optimal in space with respect to the energy norms of pressure and displacement.However,they only get the suboptimal error estimate in time,that is,the error estimate is only of order O(?t(?))in time using the backward Euler difference method.The novel weak Galerkin(WG)method was first introduced in[37,38]for second-order elliptic problems.The trial and test functions of WG method can take separate values/definitions on the interior of each element and its boundary,that is,weak functions have the form v = ?v0,vb} with v = v0 inside of the element and v?vb on the boundary of the element.Therefore,a discrete weak gradient operator arising from local RT or BDM elements is defined.Latter,the new weak Galerkin method,where a stabilization term is added,provides a convenient flexibility in both numerical approximation and mesh generat ion[3 9].The comparative study on the WG method with DG method and mixed finite element method can be found in[40].Compared to mixed finite element methods,WG methods rely on usual weak formulations,have plenty of choices for approximating finite element spaces,result in definite linear systems that are easier to solve.Compared to DG methods,WG methods do not need penalty factor,jumps and averages in discrete variational form.Then the method has been successfully applied to the parabolic problems[41,42],Helmholtz equations[43],Maxwell equations[44],Stokes equations[45],Darcy-Stokes equation[46],linear elasticity problem[47],etc.Darcy's empirical flow model represents a simple linear relationship between Darcy velocity and the gradient of pressure in a porous media[78,79].However,Darcy's law is valid when the velocity u is extremely small.In 1901,Forchheimer[80-82]observed the nonlinear relationship between the pressure and Darcy velocity for the moderate Reynolds number(Re>1 approximately).In recent years,there are extensive literatures on numerical methods for Darcy-Forchheimer equation.In[67,83]and[68],Rui proposed and analyzed a block-centered finite difference scheme and a two-grid block-centered finite difference method,respectively.Structure of Darcy-Forchheimer equations indicates that mixed finite element methods are also the commonly used numerical methods.A semidiscrete mixed element scheme for the general Forchheirmer model was analyzed by E.J.Park[84].In[85,85],Girault and Wheeler proposed a mixed element approximation for the Darcy-Forchheimer equation by using a piecewise constan-t function for velocity and a Crouzeix-Raviart element for pressure.Latter,in[87],Hilda Lopez used the same mixed element scheme to approximate Darcy-Forchheimer equation by using a piecewise constant function for velocity and a P1-conforming element for pressure.In[88],a mixed element approximation,which is different from the scheme in[85],was proposed based on the RT mixed element([89-91])or the BDM mixed element([92,93]).The resulting system is a nonlinear system when approximating Darcy-Forchheimer equation by the above mixed finite element method.Therefore,we need an iterative procedure to solve it,which leads to highly computational cost.For Maxwell's equations,Monk[137]derived the first superconvergence result in special discrete norms in 1994.Later,Brandts[122]performed another super-convergence analysis for the two-dimensional(2D)Maxwell's equations.Lin and coworkers[134,135]systematically developed some global superconvergence re-sults by using the socalled Lin's integral identity technique[136,145].Qiao et al.[139]obtained the superconvergence result for time-harmonic Maxwell's equations solved by nonconforming finite element methods on Cartesian grids with post-processing operators.Due to the unusual physical properties,metamaterial[132]is becoming a hot topic which attracts many researches.Li and coworkers first derived the superconvergence result for edge elements on rectangular and hexahedral grids[128,131].Then they extended the superconvergence result to edge elements on triangular grids and tetrahedral grids[129,130].Due to the complexity of the superconvergence analysis and implementation of edge elements,Li's works are mainly obtained for the lowest-order edge elements.However,high-order edge elements[119,121]are often used for Maxwll's equations.Basing on above background of poroelasticity,Darcy-Forchheimer model and Maxwell's equations,the dissertation respectively studies the coupled WG and mixed finite element method,fully decoupled WG method and staggered finite dif-ference method on nonuniform grids of poroealsticity problem,the two-grid stabilized mixed finite element method for Darcy-Forchheimer problem,and the super-convergence analysis of high-order rectangular edge elements for time-harmonic Maxwell's equations.In chapter 1,we propose a coupling of a WG method for the displacement of the solid phase with a standard mixed finite method for the pressure and velocity of the fluid phase.We give the poroelasticity model and the mixed variational formulation.Then we define the weak finite element space and the mixed finite element space,introduce some projection operators,present the discrete variational scheme and prove the stability,existence and uniqueness of the scheme.Subsequently,we obtain the uniform-in-time error estimates of displacement,velocity and pressure for both semidiscrete scheme and fully discrete scheme.In addition,we do not use Gronwall inequality and the assumption that c0 is uniformly positive in the error analysis.Finally,three numerical experiments are carried out to verify the accuracy and efficiency of the method.The first two examples show that we get the optimal error estimate of the uniform-in-time error estimates of displacement,velocity and pressure under both case c0>0 and c0 =0.The third example is the cantilever bracket problem.We observe that the pressure is smooth and stable using the coupled WG and mixed finite element method,whereas it is oscillating using the coupled continuous Galerkin method and mixed finite element method.This verifies that our method can efficiently eliminate the spurious pressure oscillation caused by c0= 0.In chapter 2,we firstly give the poroelasticity model and the variational formulation.We define the lowest order weak finite element space,weak gradient and weak divergence of vector and scalar function.Then we present the split-ting weak finite element scheme for the poroelasticity problem.The numerical scheme is described in two parts.Firstly,we solve for the pressure using the fluid equation,then we solve for the displacement using the mechanics equation.Therefore,the poroelasticity equation is fully decoupled.To complete the definition of the numerical scheme,we need to address the issue of approximating ph1 and uh1.So we give the initial splitting WG scheme.In the initial schemes,we eliminate the divergence term.So we need the assumption that by choosing a small initial time step?t1,the change in time of u should be small enough to be neglected.Then we prove the existence and uniqueness of the scheme and get the optimal error estimate in time and space with respect to the energy norms of displacement and pressure.The stabilization parameter in the scheme can be accurately estimated.Finally,the numerical experiments show that the energy norms of pressure and displacement in time and space are optimal for both cases c0>0 and c0 approximating 0.It is consistent with the theoretical analysis.In chapter 3,we transform the poroelasticity model to the equivalent fourfield poroelasticity equation by introducing the solid pressure and fluid velocity.Then we give some notations of the block-center finite difference method and MAC finite difference method,two approximation spaces and two equivalent discrete schemes.In the scheme,The x-component of displacement and velocity is approximated at the midpoint of vertical edges of the cell and the y-component of displacement and velocity is approximated at the midpoint of horizontal edges of the cell.The fluid pressure and solid pressure are approximated at the cell center.Hence our scheme,different from[62],reduces the number of unknowns.The stability of the scheme is proved based on LBB condition.To get the error estimate,we introduce the interpolation of displacement.When the constrained specific storage coefficient c0 is bounded below by a positive constant,we obtain the optimal error estimate for displacement and pressure in discrete Hl norm on nonuiform grid.On uniform grid,we obtain the second order superconver-gence for the pressure in discrete Hl norm and for the difference quotient of the x-component(y-component)of displacement along the x-direction(y-direction)in discrete L2 norms.The difference quotient of the x-component(y-component)of displacement along the y-direction(x-direction)converges with second order ac-curacy if the terms near the boundary are not included in the norms and converge with one and a half order accuracy otherwise.For c0 ? 0,we obtain the optimal error estimate on nonuniform grid and second order superconvergence on uniform grid for the pressure in discrete L2 norm.Moreover,the stability and error estimates are uniform with respect to Lame parameter ??(0,+?)So our scheme is free of both pressure oscillations and Poisson locking.Finally,the numerical experiments verify our theory.In chapter 4,we consider the Darcy-Forchheimer model,obtain the mixed variational form using Green formulation and give the regularity assumptions of velocity u and pressure p.Then we define the P12-P1 mixed finite element space.Obviously,the equal-order mixed finite element space pairs do not satisfy the infsup stability condition,so we introduce the pressure projection stabilization term to get the discrete variational scheme.The finite element spaces of velocity and pressure satisfy a weaker infsup condition.So we prove the existence and uniqueness of the scheme.the derivatives of |u| don't exist when the velocity u is zero,we use the first derivative of(?)to replace that of |u|.Then the two-grid stabilized finite element method is presented based on Newton correction.We firstly solve the nonlinear equation on a coarse grid and then use it to linearize the nonlinear equation on a fine grid.To obtain the error estimate,we introduce a elliptic projection,get the approximation property and prove the L2 error estimates of velocity and pressure on coarse grid.Subsequently,we obtain the L2 error estimate of two-grid method.To balance the errors caused by coarse grid,fine grid,and e,we should take e to be O(h)and H to be O(h1/2).Finally,three numerical experiments are carried out to verify the theoretical analysis.The first two examples show that the stabilized MFEM and the two-grid method keep the basically same errors and convergence rates.Furthermore,the computational complexity of two-grid algorithm is much less than that of the stabilized MFEM.The third example is the injection-production problem.The numerical solutions of two-grid method are almost same as that of stabilized MFEM.In chapter 5,we consider the time-harmonic Maxwell's equations and give some notations.We introduce the Nedelec interpolation operator and standard L2 projection operator,prove the superclose results of k-th edge elements.Then we give the superconvergence interpolation analysis of second order Nedelec element,obtain the superconvergence interpolation results of E,H,cur lE and the first and second derivatives of E.Using four Gauss points,we define the discrete l2 norms of scale and vector functions.Then we get the superconvergence rates O(h3)for ?E-Eh?l2,?H—Hh?l2 and ?curl(E—Eh)?l2 on both non-uniform and anisotropic rectangular meshes.The numerical experiments show the superconvergence rates O(h2)for the first derivatives and the second derivatives on both nonuniform and anisotropic rectangular meshes.By similar analysis,we get the superconvergence interpolation results of third order Nedelec element.Using nine Gauss points,we define the discrete l2 norms of scale and vector functions.Then we get the superconvergence rates O(h4)for E,H,curlE on both non-uniform and anisotropic rectangular meshes.Numerical results are consistent with theoretical analysis. |
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