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Adaptive Finite Element Method for Solving Poisson Partial Differential Equation

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Adaptive Finite Element Method for Solving Poisson Partial Differential Equation Journal of Nepal Mathematical Society (JNMS), Vol. 4, Issue 1(2021); K.C. Gokul, R. P. Dulal Adaptive Finite Element Method for Solving Poisson Partial Differential Equation Gokul K C1, Ram Prasad Dulal1 1 Department of Mathematics, School of Science, Kathmandu University, Dhulikhel, Nepal Correspondence to: Ram Prasad Dulal, Email: [email protected] Abstract: Poisson equation is an elliptic partial differential equation, a generalization of Laplace equation. Finite element method is a widely used method for numerically solving partial differential equations. Adap- tive finite element method distributes more mesh nodes around the area where singularity of the solution happens. In this paper, Poisson equation is solved using finite element method in a rectangular domain with Dirichlet and Neumann boundary conditions. Posteriori error analysis is used to mark the refinement area of the mesh. Dorfler adaptive algorithm is used to refine the marked mesh. The obtained results are compared with exact solutions and displayed graphically. Keywords: Poisson equation, Finite element method, Posteriori error analysis, Adaptive mesh refinement DOI: https://doi.org/10.3126/jnms.v4i1.37107 1 Introduction Differential equations arise in many areas of applications such as in science and engineering. Most of the problems in almost all the scientific and engineering areas are modeled using partial differential equations (PDEs). Partial differential equation arise when a dependent variable depends on two or more independent variables. One of the widely known method for solving PDEs is finite element method (FEM). The Finite element method is a numerical analysis technique for obtaining approximate solutions to a wide variety of engineering problems. A finite element model of a problem gives a piecewise approximation to the governing equations. The basic premise of the FEM is that a solution region can be analytically modeled or approximated by replacing it with an assemblage of discrete elements[5]. 1.1 Poisson partial differential equation Poisson equation is a partial differential equation, named after Simeon Denis Poisson. The Poisson equation is derived from the following general elliptic partial differential equation[8]: @ @u @u @ @u @u − a + a − a + a + a u − f = 0 (1) @x 11 @x 12 @y @y 21 @x 22 @y 00 For particular case, setting a11 = a22 = k(x; y) = k (constant) and a21 = a21 = a00 = 0, the equation reduces to: @2u @2u −k + = f(x; y) @x2 @y2 −kr2u = f ) −k∆u = f (2) 2 @2 @2 where, ∆ = r = @x2 + @y2 is called Laplacian operator. The equation (2) is the simplest form of elliptic PDE called Poisson PDE consisting dependent variable u and independent variables x and y. The function f(x; y) is called source function. 1 Adaptive Finite Element Method for Solving Poisson Partial Differential Equation 1.2 Adaptive finite element method (AFEM) An adaptive finite element method (AFEM) was developed in the early 1980s. The basic concept of adaptivity developed in the FEM is that, when a physical problem is analyzed using finite elements, there exists some discretization errors caused owing to the use of the finite element model. These errors are calculated in order to assess the accuracy of the solution obtained. If the errors are large, the finite element model is refined through reducing the size of elements or increasing the order of interpolation functions. The new model is re-analyzed and the errors in the new model are re-calculated. This procedure is continued until the calculated errors fall below the specified permissible values. The key features in AFEM are the estimation of discretization errors and the refinement of finite element models[7]. It minimizes the errors by the concept of posteriori error analysis and using adaptive algorithm. It can be shown in the following diagram[7]: Figure 1: Adaptive finite element process 2 Methodology 2.1 Weak form of poisson equation In the development of the weak form, we consider a typical triangular element Ωe of the finite element mesh and develop the finite element model for the Poisson PDE (2). Multiplying (2) by an arbitrary test 1 function w = w(x; y) 2 H0 (Ωe), which is assumed to be differentiable with respect to x and y, we get weak form of (2) as[4]: Z @w @u @w @u Z I k + k dxdy = wfdxdy + wqnds (3) Ωe @x @x @y @y Ωe Γe The equation (3) can be written as: Be(w; u) = Le(w) (4) where, Z @w @u @w @u k + k dxdy = Be(w; u) Ωe @x @x @y @y Z I e wfdxdy + wqnds = L (w) Ωe Γe 2.2 Finite element solution process 2.2.1 Interpolation functions The governing equation (2) is equivalent to the weak form of equation (4) over the domain Ωe and also contains natural boundary conditions. The weak form requires the approximation chosen for u should be at least linear in both x and y. There are no terms that are identically zero. Since the primary variable is a function itself, the Lagrange family of interpolation functions is admissible. Let u is the approximated solution over a typical element Ωe defined by the expression[8]: n e X e e u = u(x; y) ≈ uh(x; y) = uj j (x; y): (5) j=1 2 Journal of Nepal Mathematical Society (JNMS), Vol. 4, Issue 1(2021); K.C. Gokul, R. P. Dulal e e th e Where, uj is the value of uh at the j node (xj; yj) of the element and j is the Lagrange interpolation function with the property that: e i (xj; yj) = δij; i = 1; 2; :::; n (6) Putting the approximated value of (5) in (4), we get: n Z e e Z I X @w @ j @w @ j k + k dxdy ue = wfdxdy + wq ds (7) @x @x @y @y j n j=1 Ωe Ωe Γe This equation must hold for every admissible choice of weight function w. Since we need n independent e e e e algebraic equations to solve for n unknowns (u1; u2; u3; :::; un), we choose n linearly independent functions e e e e for w. The functions of w are w = ( 1; 2; 3; :::; n). Hence we write: e e e e e u = u1 u2 u3 : : : un e e e e T w = 1 2 3 : : : n = This particular choice of weight function is a natural one when the weight function is viewed as a virtual Pn variation of the dependent unknowns (w = @u = i=1 @ui i) and the resulting finite element model is called weak form finite element model[8]. e e e e For each choice of w, we obtain an algebraic relation among (u1; u2; u3; :::; un). We label the algebraic equation resulting from substituting successively the following in (7): e e e e w = 1; w = 2; w = 3; :::; w = n: to get n algebraic equations. Hence the general ith form of the system can be written as: n Z e e e e Z I X @ @ j @ @ j i k + i k dxdy ue = f edxdy + q eds where @x @x @y @y j i n i j=1 Ωe Ωe Γe Z e e e e Z I e @ i @ j @ i @ j e e e e Kij = k + k dxdy; fi = f i dxdy; Qi = qn i ds Ωe @x @x @y @y Ωe Γe Which reduces to: n X e e e e Kijuj = fi + Qi ; i = 1; 2; 3; :::; n (8) j=1 e e e e e The term Kij is a bilinear form and is denoted as Kij = B ( i ; j ). Equation 8 abbreviated in vector form as: Keue = f e + Qe (9) The matrix Ke is called the coefficient matrix and the vector f e is called the source vector. The weak form e and the finite element matrices in (9) shows that i must be at least linear functions of x and y. The complete linear polynomial in x and y in Ωe is of the form: e e e e uh(x; y) = c1 + c2x + c3y (10) e where, ci 's are constants for i = 1; 2; 3: The set f1; x; yg is linearly independent and complete. The e e equation (10) defines a unique plane for fixed ci . In particular, uh(x; y) is uniquely defined on a triangle. Let (x1; y1); (x2; y2) and (x3; y3) be the vertices of the triangle such that: e e e e e e uh(x1; y1) = u1; uh(x2; y2) = u2; uh(x3; y3) = u3: (11) e e The constants ci ,(i = 1; 2; 3) in (10) can be expressed in terms of three nodal values ui (i = 1; 2; 3). The particular nodes of the triangle can be expressed as: e e e e e uh(x1; y1) ≡ u1 = c1 + c2x1 + c3y1 e e e e e uh(x2; y2) ≡ u2 = c1 + c2x2 + c3y2 (12) e e e e e uh(x3; y3) ≡ u3 = c1 + c2x3 + c3y3: 3 Adaptive Finite Element Method for Solving Poisson Partial Differential Equation The matrix form of this system of equations is: 2 3 2 e3 2 e3 1 x1 y1 c1 u1 e e 41 x2 y25 4c25 = 4u25 (13) e e 1 x3 y3 c3 u3 where, A is coefficient matrix, C is constant matrix and ue is solution matrix such that: 2 3 2 e3 2 e3 1 x1 y1 c1 u1 e e e A = 41 x2 y25 ; C = 4c25 ; u = 4u25 : e e 1 x3 y3 c3 u3 The inverse matrix of A is given by: 2α α α 3 1 1 2 3 A−1 = : β β β (14) jAj 4 1 2 3 5 γ1 γ2 γ3 where αi; βi; γi are constants depend only on the global coordinates of element nodes (xi; yi) s.t.: αi = xjyk − xkyj; βi = yj − yk; γi = xk − xj (15) for i 6= j 6= k and i; j; k permute in a natural order.
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