In order to solve some complicated problems involving partial differential equations (PDEs) or systems of them, several numerical methods have been exploited. Among a variety of existing works, the finite difference, finite volume, and finite element are more outstanding and applicable. The discontinuous Galerkin (DG) method is a class of finite element methods with discontinuous piecewise polynomial space that has many flexibilities because of producing discontinuous approximate solutions. However, due to some limitations of the traditional DG method, e.g. existing inconsistency in numerical solutions, the local DG (LDG) method is invented which can be applied to solve problems with higher-order space derivatives. The framework of the LDG method is based on reformulating a higherorder linear/nonlinear PDE to a first-order system of PDEs. Classical DG methods preserve high-order accuracy for both the solution and its derivatives but some continuous Galerkin (CG) methods such as the well-known finite element method are more economical in particular in solving some steady-state problems. Afterwards, some HDG methods have been developed for keeping the high-order accuracy of DG methods and improving their efficiencies. Due to the discontinuity in the HDG solutions and the way of defining numerical fluxes, the HDG method is known as an outstanding method with a lot of certain priority and flexibility. On the other hand, an HDG method exploits CG solvers simultaneously for increasing the performance of the method. Moreover, the HDG method is actually quite computationally competitive with CG methods in solving steady-state problems at high orders. In fact, This method was initially developed in order to address the large number of degrees of freedom that more standard DG methods display for steady-state problems, or, for that matter, any system requiring a global solve. Discontinuous bases in the DG method cause discontinuous solutions along elements’ edges. So we have multivalued function eva‎luations at inner-element fluxes. This increases the degree of freedom. In contrast, HDG methods need less number of globally coupled degrees of freedom rather than DG methods. The first step of the HDG method is defining the auxiliary variables to form the first-order system of equations. Then, numerical fluxes are introduced inside each element in terms of the numerical traces and stabilization parameters (the stability analysis determine sufficient condition on stabilization parameters to construct a stable method). Numerical traces as global unknowns, are assumed to be single-valued functions on each face and depend only on the number of elements for an arbitrary polynomial. Next, the conservation of numerical fluxes is imposed to get extra sets of equations. These equations are expressed in terms of the increments of approximate traces in each iteration. Then, the local approximate solutions will be obtained, by solving these global equations and substituting the numerical traces back into local equations. In this thesis, we try to construct some stable schemes for solving some PDEs using the HDG method. Firstly, some explanations of prerequisites such as Sobolev space, Hilbert space, projection operators, mesh generation, and approximate spaces are presented. In summary, for discretizing spatial domain Ω, we use uniform meshes (for dimension one) and triangular or rectangular meshes (for dimension greater than one). Also, discontinuous finite element space and skeleton space as subspaces of broken Sobolev space, are our approximate spaces. After stating the preliminaries, the first part of our research work starts and the HDG method is employed for solving some different types of evolution equations.

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حميدرضا مرزبان , هادي روحاني , عليرضا سهيلي