Mixed Finite Volume Methods Introduction

Consider the following second-order variable-coefficient elliptic boundary value problem on a bounded polygonal domainΩin R² with the boundary (?)Ω:Where the coefficient K is a symmetric and uniformly positive-definite matrix, i.e., the following relation holds:(or ), whereα₁,α₂ are two positive constants. We introduce a new variable , and rewrite (1) as the following system of first order partial differential equations,When the right hand side f, the diffusion tensor K and the domainΩ, are such that the solution is smooth enough, e.g., u∈(H¹(Ω))², p∈H²(Ω)∩H₀¹(Ω) (or u∈H₀¹(Ω),p∈H²(Ω)∩H¹(Ω)), then the problem (1) and (3) are equivalent. This method is the so-called mixed method.In the mathematical modeling of fluid flow in porous media, u and p represent the velocity and pressure fields, respectively, and K is the diffusion tensor. The first equation represents conservation of mass, and the second equation is the Darcy law. If we choose the function spaces as:Then the associated weak formulation for the system (3) is to find (u,p)∈V×W such that,Where, (·,·) denotes the standard inner product in L²(Ω) or (L²(Ω))². Next, we consider the finite volume methods for the problem (3), i.e., the mixed finite volume methods.First, our consideration goes down to the primal triangulation case and the primal quadrilateral grid case in order to introduce the finite volume method with two partitions (primal and dual partitions).First, we consider the partition of the domain. As for the first case (see the bottom figure in picture 2.1), let T_h, denote the primal partition, and the covolume associated with the edge E of the primal partition is: Q_E = T_E⁺∪E∪T_E^- (particularly, on the boundary, it becomes to be Q_E = T_E⁺, or Q_E = T_E^-). So a dual partition (a union of covolums) are: Q_h = {Q_E}; For the second case (see picture 2.2 and picture 2.3), let Q_h represent the primal partition, then the left-hand quadrilateral between the edges e_ij⁽²⁾ and e_i+1/2,j and the right-hand quadrilateral between the edges e_i+1,j⁽²⁾ and e_i+1/2,j form the volume Q_i+1/2,j ( the same with Q_i,j+1/2 on the other direction ) together. Where, e_i+1/2,j denotes a common edge of Q_ij and Q_i+1,j. e_ij⁽²⁾ (resp., e_ij⁽¹⁾ on the other direction) represents the line segment joining the midpoints of the edges of a primal element.Now we consider the function spaces. We choose the space of piecewise constant functions with respect to the primal partition T_h as the pressure space W_h (?) W; And choose the lowest order Raviart-Thomas space as the velocity space V_h (?) V. So for the primal triangulation, the velocity can be characterized by the piecewise polynomials of the form [a₁ + cx, b₁ +cy]^T that have continuous normal traces across the edges of the triangles T∈T_h, as for the primal quadrilateral grid, it can be represented as [a₁ + cx, b₁ + dy]^T. In order to define the test function space, we use a injective, bounded linear operatorγ_h, with which we relate it to the finite element methods. We choose the test function space as: Y_h =γ_hV_h. Then the mixed finite volume methods corresponding to the problem (1) can be written as the following unified formulation:Find (u_h, p_h)∈V_h×W_h, such that,where,as for the primal quadrilateral grid case, b(γ_hv_h,p_h) = -div(v_h,p_h).Next, we consider the mixed finite volume methods with a single grid. And we still consider the two cases of triangulation and quadrilateral grid.When the partition is a triangulation, we choose the nonconforming P₁ space Y_h,0 as our pressure space; And as the case of the quadrilateral grid, we choose the rotated-Q1 nonconforming space N_h as the pressure space. As for the velocity space, no matter it is a triangulation or a quadrilateral grid, we still choose the lowest order Raviart-Thomas space V'_h(?) H(div;Ω) (see the covolume method on two partitions). Now we consider the test function space for each grid respectively. For the triangulation case, we choose the characteristic function of each triangular element X_K, and the vector function (-|X)_K (any constant vector multiple of X_K) as our test functions; But for the quadrilateral case, in order to make sure the unknowns are equal to the numbers of the equations, we choose a vector function (?)X(X∈N_h(K)) from the function space of three dimension (?)N_h(K), together with the character function of each element as the test functions. Then the finite volume method can be written as follows:Find (u_h, p_h)∈V'_h×Y_h,0(N_h), such that,It can be proved that this finite volume method is in fact equal to a nonconforming finite element method for the pressure p with a local recovery of the flux.The above method is different from the mixed covolume method with a dual partition, with which we seek the pressure solution on the primal grid, but calculate the velocity on the dual grid. Since there is only a single partition, it is easier for both the calculation and the conservation of the datum. And it is rather surprising that it achieves stability using a single grid system while violating the inf-sup condition on the pressure-velocity spaces.