An Introduction to the Finite Element Method for Differential Equations. Mohammad Asadzadeh

      1 I) The constant coefficients caseA second‐order PDE in two dimensions, with constant coefficients, can be written in its general form asWe introduce the discriminant : a quantity that specifies the role of the coefficients of the second‐order terms, in determining the equation type in the sense that the equation isElliptic: if Parabolic: if and Hyperbolic: if Example 1.2Below are the classes of the most common, second‐order differential equations (in any dimension) and examples of their forms in :Potential equationHeat equationWave equation (elliptic) (parabolic) (hyperbolic).

      2 II) The case of variable coefficientsIn the variable coefficients case, one can only have a local classification.Example 1.3Consider the Tricomi equation of gas dynamics:Here, the coefficient is not a constant, and we have and . Hence, and consequently, the domain of ellipticity is , and so on, see Figure 1.1.Figure 1.1 Tricomi equation: an example of a variable coefficient classification.

      1 Initial value problems (IVPs)The simplest differential equation is for . But for any such , also for any constant . To determine a unique solution, a specification of the initial value is generally required. For example for , we have and the general solution is . With an initial value of , we get . Hence, the unique solution to this IVP is . Likewise, for a time‐dependent differential equation of second order (two time derivatives), the initial values for , i.e. and , are generally required. For a PDE such as the heat equation, the initial value can be a function of the space variable.Example 1.4The wave equation, on the real line, augmented with the given initial data:Remark 1.1Note that, here, for the unbounded spatial domain , it is required that (or =constant) as . This corresponds to two boundary conditions.

      2 Boundary value problems (BVP)Example 1.5 (A BVP in )Consider the one‐dimensional stationary heat equation:In order to determine a solution uniquely (see Remark 1.2), the differential equation is complemented by boundary conditions imposed at the boundary points and ; for example and , where and are given real numbers.Example 1.6(A BVP in ).The Laplace equation in ,Remark 1.2In general, in order to obtain a unique solution for a (partial) differential equation, one needs to supply physically adequate boundary data. For instance for the potential problem , stated in a rectangular domain , to determine a unique solution we need to give two boundary conditions in the ‐direction, i.e. the numerical values for and , and another two in the ‐direction: the numerical values of and . Whereas to determine a unique solution for the wave equation , it is necessary to supply two initial conditions in the time variable , and two boundary conditions in the space variable . We observe that in order to obtain a unique solution, we need to supply the same number of boundary (initial) conditions as the order of the differential equation in each spatial (time) variable. The general rule is that one should supply as many conditions as the highest order of the derivative in each variable. See also Remark 1.1, in the case of unbounded spatial domain.

      3 Eigenvalue problems (EVPs)Let be a given square, say matrix. The relation , is a linear equation system, where is an eigenvalue and is an eigenvector. In the Example 1.7 below, we introduce the corresponding terminology for differential equations.Example 1.7A differential equation describing a time‐independent vibrating string is given bywhere is an eigenvalue and is an eigenfunction. and are boundary values.To derive this equation, we consider the differential equation for a time‐dependent vibrating string, with small displacement , which is fixed at the end points, viz.Then, using the separation of variables technique, this equation is split into two EVPs: Insert (a nontrivial solution) into the differential equation to get(1.2.4) Dividing (1.2.4) by separates the variables so that we have(1.2.5) Consequently we get two ODEs (two EVPs):(1.2.6)

      1 Dirichlet boundary conditionHere, the solution is known at the boundary of the domain, asThis is the case, for example, describing a fixed temperature at the boundary.

      2 Neumann boundary conditionIn this case, the derivative of the solution in certain direction is given:,where is, e.g. the outward unit normal to at and

      3 Robin boundary condition (a combination of 1 and 2),In a homogeneous case, i.e. for , Robin condition means that the heat flux through the boundary is proportional to the temperature (in fact the temperature difference between the inside and outside) at the boundary.Example 1.8In two dimensions, with and hence, for , we have , see Figure 1.2.Figure 1.2 Outward unit normal at a point .Example 1.9Let . We determine the normal derivative of in (the assumed normal) direction . The gradient of is the vector valued function , where and are the unit orthonormal basis in : and . Note that is not a unit vector. The normalized is obtained as , i.e.Thus, which givesThe usual three criteria to deal with1) In theoryA given differential equation problem is called well‐posed if the following three conditions hold true:I1. Existence: there exists at least one solution .I2. Uniqueness: we have either one solution or no solutions at all.I3. Stability: the solution depends continuously on the data.Note: A property that concerns behavior, such as growth or decay, or perturbations of a solution as time increases is generally called a stability property.2) In applicationsII1. Construction: exact design of the solution.II2. Regularity: how smooth is the found solution.II3. Approximation: when an exact construction is not possible.

      1 Separation of variables method: The separation of variables technique reduces the (PDEs) to simpler EVPs (ODEs). This method is known as Fourier method, or solution by eigenfunction expansion.

      2 Variational formulation method: Variational formulation or the multiplier method is a strategy for extracting information by multiplying a differential equation by suitable test functions and then integrating (e.g. like Fourier or Laplace transforms). This is also referred to as The Energy Method. A discrete version of this is the subject of our study.

      3 Green's function method: Fundamental solutions, or solution of integral equations, which we have briefly addressed in the chapter of mathematical tools is the subject of an advanced PDE course.