A posteriori estimates for partial differential equations
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In addition to Eq. Such knowledge can be applied in the initial condition and boundary conditions for Eq. In many situations, PDEs cannot be resolved with analytical methods to give the value of the dependent variables at different times and positions. It may, for example, be very difficult or impossible to obtain an analytic expression such as:.
Rather than solving PDEs analytically, an alternative option is to search for approximate numerical solutions to solve the numerical model equations. The finite element method is exactly this type of method — a numerical method for the solution of PDEs. Similar to the thermal energy conservation referenced above, it is possible to derive the equations for the conservation of momentum and mass that form the basis for fluid dynamics. Further, the equations for electromagnetic fields and fluxes can be derived for space- and time-dependent problems, forming systems of PDEs.
Continuing this discussion, let's see how the so-called weak formulation can be derived from the PDEs. Assume that the temperature distribution in a heat sink is being studied, given by Eq. The boundary conditions at these boundaries then become:.
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The outward unit normal vector to the boundary surface is denoted by n. Equations 10 to 13 describe the mathematical model for the heat sink, as shown below. The domain equation and boundary conditions for a mathematical model of a heat sink.
The next step is to multiply both sides of Eq. A Hilbert space is an infinite-dimensional function space with functions of specific properties. It can be viewed as a collection of functions with certain nice properties, such that these functions can be conveniently manipulated in the same way as ordinary vectors in a vector space. For example, you can form linear combinations of functions in this collection the functions have a well-defined length referred to as norm and you can measure the angle between the functions, just like Euclidean vectors. Indeed, after applying the finite element method on these functions, they are simply converted to ordinary vectors.
The finite element method is a systematic way to convert the functions in an infinite dimensional function space to first functions in a finite dimensional function space and then finally ordinary vectors in a vector space that are tractable with numerical methods. The weak formulation is obtained by requiring 14 to hold for all test functions in the test function space instead of Eq. A problem formulation based on Eq. In the so-called Galerkin method , it is assumed that the solution T belongs to the same Hilbert space as the test functions.
The weak formulation, or variational formulation , of Eq. The relations in 14 and 15 instead only require equality in an integral sense. For example, a discontinuity of a first derivative for the solution is perfectly allowed by the weak formulation since it does not hinder integration. It does, however, introduce a distribution for the second derivative that is not a function in the ordinary sense. As such, the requirement 10 does not make sense at the point of the discontinuity.
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A distribution can sometimes be integrated, making 14 well defined. It is possible to show that the weak formulation, together with boundary conditions 11 through 13 , is directly related to the solution from the pointwise formulation. And, for cases where the solution is differentiable enough i.
This is the first step in the finite element formulation. With the weak formulation, it is possible to discretize the mathematical model equations to obtain the numerical model equations. The Galerkin method — one of the many possible finite element method formulations — can be used for discretization.
First, the discretization implies looking for an approximate solution to Eq. The discretized version of Eq. The unknowns here are the coefficients T i in the approximation of the function T x.
From Eq. Once the system is discretized and the boundary conditions are imposed, a system of equations is obtained according to the following expression:. The right-hand side is a vector of the dimension 1 to n. If the source function is nonlinear with respect to temperature or if the heat transfer coefficient depends on temperature, then the equation system is also nonlinear and the vector b becomes a nonlinear function of the unknown coefficients T i.
One of the benefits of the finite element method is its ability to select test and basis functions.
Forschung – Department Mathematik
It is possible to select test and basis functions that are supported over a very small geometrical region. This implies that the integrals in Eq. The support of the test and basis functions is difficult to depict in 3D, but the 2D analogy can be visualized. Assume that there is a 2D geometrical domain and that linear functions of x and y are selected, each with a value of 1 at a point i , but zero at other points k. The next step is to discretize the 2D domain using triangles and depict how two basis functions test or shape functions could appear for two neighboring nodes i and j in a triangular mesh.
Two neighboring basis functions share two triangular elements.
A posteriori error analysis and adaptive methods for partial differential equations
As such, there is some overlap between the two basis functions, as shown above. These contributions form the coefficients for the unknown vector T that correspond to the diagonal components of the system matrix A jj. Say the two basis functions are now a little further apart. These functions do not share elements but they have one element vertex in common. As the figure below indicates, they do not overlap. When the basis functions overlap, the integrals in Eq. When there is no overlap, the integrals are zero and the contribution to the system matrix is therefore zero as well.
This means that each equation in the system of equations for 17 for the nodes 1 to n only gets a few nonzero terms from neighboring nodes that share the same element.
Parameter Estimation of Partial Differential Equation Models.
The system matrix A in Eq. The solution of the system of algebraic equations gives an approximation of the solution to the PDE. The denser the mesh, the closer the approximate solution gets to the actual solution.
The thermal energy balance in the heat sink can be further defined for time-dependent cases. Here, the coefficients T i are time-dependent functions while the basis and test functions depend just on spatial coordinates. Further, the time derivative is not discretized in the time domain. One approach would be to use FEM for the time domain as well, but this can be rather computationally expensive. Alternatively, an independent discretization of the time domain is often applied using the method of lines.
For example, it is possible to use the finite difference method.
In its simplest form, this can be expressed with the following difference approximation:. Two potential finite difference approximations of the problem in Eq.