Several Studies On Finite Volume Element Methods On General Quadrilateral Meshes

Finite volume element methods are an important class of numerical methods for solving partial differential equations and are widely used in many fields of science and engineering.The main advantage of these methods is that they maintain the local conservation of physical quantities,and they are also easy to handle complex region boundaries and improve numerical accuracy.However,because the space of test functions is different from the space of trial functions in the finite volume element method,its theoretical analysis is more complicated than the corresponding finite element method,especially when the mesh transformation is not a simple affine transformation or when the problem to be solved is nonlinear and time-dependent.Therefore,in this paper,effective finite volume element schemes are constructed on general quadrilateral meshes for several typical model problems,and rigorous theoretical analyses are carried out.Specific research contents include:In the first part,for second-order nonlinear elliptic equations,a finite volume method scheme is constructed on quadrilateral grids and corresponding theoretical analyses are provided.The calculation domain is first divided into general convex quadrilateral grids and their corresponding average-centered dual grids.The bilinear element space and piecewise constant function space are selected as the trial function space and test function space respectively to construct the corresponding finite volume method scheme.Secondly,under the condition of h2 parallelogram grid,the boundedness and coerciveness of the bilinear form are proved,and the existence and uniqueness of the numerical solution is given by utilizing the Brouwer fixed point theorem in a bounded subspace of the bilinear element space.Finally,under certain regularity assumptions,the optimal order estimates of ‖▽(u-uh)‖Lq(q≥2)and ‖u-uh‖0 are established,and the theoretical results are verified by numerical experiments.In the second part,a semi-discrete scheme and a backward Euler fully discrete scheme are constructed for a two-dimensional nonlinear parabolic equation on h2-parallelogram grids.By utilizing the elliptic projection operator and the quasi-symmetry of the bilinear form(·,∏h*·),error estimates of the discrete solution measured in H1 and L2 norms are established respectively.The existence of the discrete solution is proven by using the Brouwer fixed point theorem in a bounded subset of the trial function space,with the aid of an appropriate nonlinear mapping and the corresponding theoretical results for nonlinear elliptic problems.In addition,in order to improve computational efficiency,a linearized discrete format is designed by taking the unknown function in the nonlinear coefficient as the previous time layer,and the convergence of this scheme is proved by measuring it in H1 and L2 norms.Several numerical examples are presented to verify the effectiveness of the scheme and the correctness of the theoretical results.In the third part,two isoparametric bilinear finite volume element schemes are established for second-order linear elliptic problems on non-matching h2-parallelogram grids:the first scheme has only one unknown variable at each node(including hanging nodes),and the second scheme has the same number of unknown variables at each node as the number of dual sub-regions at this node.Under the condition of h2-parallelogram grid,the boundedness and coerciveness of the bilinear form are proven,and the a priori estimates of the numerical solution in terms of the grid-dependent norm are given.A residual-based posterior error estimate indicator is designed,and the rigorous upper and lower bounds of the a posteriori error are estimated.Several typical numerical experiments are conducted to verify the effectiveness of the method and the correctness of the theoretical analysis.